Basic Forms ∫ ( a f ( x ) + b g ( x ) ) d x = a ∫ f ( x ) d x + b ∫ g ( x ) d x \int{\left({a}{f{{\left({x}\right)}}}+{b}{g{{\left({x}\right)}}}\right)}{d}{x}={a}\int{f{{\left({x}\right)}}}{d}{x}+{b}\int{g{{\left({x}\right)}}}{d}{x} ∫ ( a f ( x ) + b g ( x ) ) d x = a ∫ f ( x ) d x + b ∫ g ( x ) d x a {a} a b {b} b ∫ u d v = u v − ∫ v d u \int{u}{d}{v}={u}{v}-\int{v}{d}{u} ∫ u d v = u v − ∫ v d u ∫ x n d x = { x n + 1 n + 1 + C if n ≠ − 1 ln ∣ x ∣ + C if n = − 1 \int{{x}}^{{n}}{d}{x}={\left\{\begin{array}{c}\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}{\quad\text{if}\quad}{n}\ne-{1}\\{\ln}{\left|{x}\right|}+{C}{\quad\text{if}\quad}{n}=-{1}\\ \end{array}\right.} ∫ x n d x = { n + 1 x n + 1 + C if n = − 1 ln ∣ x ∣ + C if n = − 1 ∫ a x d x = a x ln ( a ) + C \int{{a}}^{{x}}{d}{x}=\frac{{{{a}}^{{x}}}}{{{\ln{{\left({a}\right)}}}}}+{C} ∫ a x d x = l n ( a ) a x + C ∫ e x d x = e x + C \int{{e}}^{{x}}{d}{x}={{e}}^{{x}}+{C} ∫ e x d x = e x + C ∫ sin ( x ) d x = − cos ( x ) + C \int{\sin{{\left({x}\right)}}}{d}{x}=-{\cos{{\left({x}\right)}}}+{C} ∫ sin ( x ) d x = − cos ( x ) + C ∫ cos ( x ) d x = sin ( x ) + C \int{\cos{{\left({x}\right)}}}{d}{x}={\sin{{\left({x}\right)}}}+{C} ∫ cos ( x ) d x = sin ( x ) + C ∫ sec 2 ( x ) d x = tan ( x ) + C \int{{\sec}}^{{2}}{\left({x}\right)}{d}{x}={\tan{{\left({x}\right)}}}+{C} ∫ sec 2 ( x ) d x = tan ( x ) + C ∫ csc 2 ( x ) d x = − cot ( x ) + C \int{{\csc}}^{{2}}{\left({x}\right)}{d}{x}=-{\cot{{\left({x}\right)}}}+{C} ∫ csc 2 ( x ) d x = − cot ( x ) + C ∫ sec ( x ) tan ( x ) d x = sec ( x ) + C \int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}={\sec{{\left({x}\right)}}}+{C} ∫ sec ( x ) tan ( x ) d x = sec ( x ) + C ∫ csc ( x ) cot ( x ) d x = − csc ( x ) + C \int{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}{d}{x}=-{\csc{{\left({x}\right)}}}+{C} ∫ csc ( x ) cot ( x ) d x = − csc ( x ) + C ∫ tan ( x ) d x = − ln ∣ cos ( x ) ∣ + C = ln ∣ sec ( x ) ∣ + C \int{\tan{{\left({x}\right)}}}{d}{x}=-{\ln}{\left|{\cos{{\left({x}\right)}}}\right|}+{C}={\ln}{\left|{\sec{{\left({x}\right)}}}\right|}+{C} ∫ tan ( x ) d x = − ln ∣ cos ( x ) ∣ + C = ln ∣ sec ( x ) ∣ + C ∫ cot ( x ) d x = ln ∣ sin ( x ) ∣ + C \int{\cot{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sin{{\left({x}\right)}}}\right|}+{C} ∫ cot ( x ) d x = ln ∣ sin ( x ) ∣ + C ∫ sec ( x ) d x = ln ∣ sec ( x ) + tan ( x ) ∣ + C \int{\sec{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sec{{\left({x}\right)}}}+{\tan{{\left({x}\right)}}}\right|}+{C} ∫ sec ( x ) d x = ln ∣ sec ( x ) + tan ( x ) ∣ + C ∫ csc ( x ) d x = ln ∣ csc ( x ) − cot ( x ) ∣ + C \int{\csc{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\csc{{\left({x}\right)}}}-{\cot{{\left({x}\right)}}}\right|}+{C} ∫ csc ( x ) d x = ln ∣ csc ( x ) − cot ( x ) ∣ + C ∫ d x a 2 − x 2 = arcsin ( x a ) + C \int\frac{{{d}{x}}}{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}={\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ a 2 − x 2 d x = arcsin ( a x ) + C ∫ d x a 2 + x 2 = 1 a arctan ( x a ) + C \int\frac{{{d}{x}}}{{{{a}}^{{2}}+{{x}}^{{2}}}}=\frac{{1}}{{a}}{\operatorname{arctan}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ a 2 + x 2 d x = a 1 arctan ( a x ) + C ∫ d x x x 2 − a 2 = 1 a arcsec ( x a ) + C \int\frac{{{d}{x}}}{{{x}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}=\frac{{1}}{{a}}\text{arcsec}{\left(\frac{{x}}{{a}}\right)}+{C} ∫ x x 2 − a 2 d x = a 1 arcsec ( a x ) + C ∫ d x a 2 − x 2 = 1 2 a ln ∣ x + a x − a ∣ + C \int\frac{{{d}{x}}}{{{{a}}^{{2}}-{{x}}^{{2}}}}=\frac{{1}}{{{2}{a}}}{\ln}{\left|\frac{{{x}+{a}}}{{{x}-{a}}}\right|}+{C} ∫ a 2 − x 2 d x = 2 a 1 ln ∣ ∣ x − a x + a ∣ ∣ + C ∫ d x x 2 − a 2 = 1 2 a ln ∣ x − a x + a ∣ + C \int\frac{{{d}{x}}}{{{{x}}^{{2}}-{{a}}^{{2}}}}=\frac{{1}}{{{2}{a}}}{\ln}{\left|\frac{{{x}-{a}}}{{{x}+{a}}}\right|}+{C} ∫ x 2 − a 2 d x = 2 a 1 ln ∣ ∣ x + a x − a ∣ ∣ + C Exponential and Logarithmic Forms ∫ x e a x d x = 1 a 2 ( a x − 1 ) e a x + C \int{x}{{e}}^{{{a}{x}}}{d}{x}=\frac{{1}}{{{{a}}^{{2}}}}{\left({a}{x}-{1}\right)}{{e}}^{{{a}{x}}}+{C} ∫ x e a x d x = a 2 1 ( a x − 1 ) e a x + C ∫ x n e a x d x = 1 a x n e a x − n a ∫ x n − 1 e a x d x \int{{x}}^{{n}}{{e}}^{{{a}{x}}}{d}{x}=\frac{{1}}{{a}}{{x}}^{{n}}{{e}}^{{{a}{x}}}-\frac{{n}}{{a}}\int{{x}}^{{{n}-{1}}}{{e}}^{{{a}{x}}}{d}{x} ∫ x n e a x d x = a 1 x n e a x − a n ∫ x n − 1 e a x d x ∫ e a x sin ( b x ) d x = e a x a 2 + b 2 ( asin ( b x ) − b cos ( b x ) ) + C \int{{e}}^{{{a}{x}}}{\sin{{\left({b}{x}\right)}}}{d}{x}=\frac{{{{e}}^{{{a}{x}}}}}{{{{a}}^{{2}}+{{b}}^{{2}}}}{\left({\operatorname{asin}{{\left({b}{x}\right)}}}-{b}{\cos{{\left({b}{x}\right)}}}\right)}+{C} ∫ e a x sin ( b x ) d x = a 2 + b 2 e a x ( asin ( b x ) − b cos ( b x ) ) + C ∫ e a x cos ( b x ) d x = e a x a 2 + b 2 ( acos ( b x ) + b sin ( b x ) ) + C \int{{e}}^{{{a}{x}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=\frac{{{{e}}^{{{a}{x}}}}}{{{{a}}^{{2}}+{{b}}^{{2}}}}{\left({\operatorname{acos}{{\left({b}{x}\right)}}}+{b}{\sin{{\left({b}{x}\right)}}}\right)}+{C} ∫ e a x cos ( b x ) d x = a 2 + b 2 e a x ( acos ( b x ) + b sin ( b x ) ) + C ∫ ln ( x ) d x = x ( ln ( x ) − 1 ) + C \int{\ln{{\left({x}\right)}}}{d}{x}={x}{\left({\ln{{\left({x}\right)}}}-{1}\right)}+{C} ∫ ln ( x ) d x = x ( ln ( x ) − 1 ) + C ∫ x n ln ( x ) d x = x n + 1 ( n + 1 ) 2 ( ( n + 1 ) ln ( x ) − 1 ) + C \int{{x}}^{{n}}{\ln{{\left({x}\right)}}}{d}{x}=\frac{{{{x}}^{{{n}+{1}}}}}{{{{\left({n}+{1}\right)}}^{{2}}}}{\left({\left({n}+{1}\right)}{\ln{{\left({x}\right)}}}-{1}\right)}+{C} ∫ x n ln ( x ) d x = ( n + 1 ) 2 x n + 1 ( ( n + 1 ) ln ( x ) − 1 ) + C ∫ 1 x ln ( x ) d x = ln ∣ ln ( u ) ∣ + C \int\frac{{1}}{{{x}{\ln{{\left({x}\right)}}}}}{d}{x}={\ln}{\left|{\ln{{\left({u}\right)}}}\right|}+{C} ∫ x l n ( x ) 1 d x = ln ∣ ln ( u ) ∣ + C Trigonometric Forms ∫ sin 2 ( x ) d x = 1 2 x − 1 4 sin ( 2 x ) + C \int{{\sin}}^{{2}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{x}-\frac{{1}}{{4}}{\sin{{\left({2}{x}\right)}}}+{C} ∫ sin 2 ( x ) d x = 2 1 x − 4 1 sin ( 2 x ) + C ∫ cos 2 ( x ) d x = 1 2 x + 1 4 sin ( 2 x ) + C \int{{\cos}}^{{2}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{x}+\frac{{1}}{{4}}{\sin{{\left({2}{x}\right)}}}+{C} ∫ cos 2 ( x ) d x = 2 1 x + 4 1 sin ( 2 x ) + C ∫ tan 2 ( x ) d x = tan ( x ) − x + C \int{{\tan}}^{{2}}{\left({x}\right)}{d}{x}={\tan{{\left({x}\right)}}}-{x}+{C} ∫ tan 2 ( x ) d x = tan ( x ) − x + C ∫ cot 2 ( x ) d x = − cot ( x ) − x + C \int{{\cot}}^{{2}}{\left({x}\right)}{d}{x}=-{\cot{{\left({x}\right)}}}-{x}+{C} ∫ cot 2 ( x ) d x = − cot ( x ) − x + C ∫ sin 3 ( x ) d x = − 1 3 ( 2 + sin 2 ( x ) ) cos ( x ) + C \int{{\sin}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{3}}{\left({2}+{{\sin}}^{{2}}{\left({x}\right)}\right)}{\cos{{\left({x}\right)}}}+{C} ∫ sin 3 ( x ) d x = − 3 1 ( 2 + sin 2 ( x ) ) cos ( x ) + C ∫ cos 3 ( x ) d x = 1 3 ( 2 + cos 2 ( x ) ) sin ( x ) + C \int{{\cos}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{3}}{\left({2}+{{\cos}}^{{2}}{\left({x}\right)}\right)}{\sin{{\left({x}\right)}}}+{C} ∫ cos 3 ( x ) d x = 3 1 ( 2 + cos 2 ( x ) ) sin ( x ) + C ∫ tan 3 ( x ) d x = 1 2 tan 2 ( x ) + ln ∣ cos ( x ) ∣ + C \int{{\tan}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{{\tan}}^{{2}}{\left({x}\right)}+{\ln}{\left|{\cos{{\left({x}\right)}}}\right|}+{C} ∫ tan 3 ( x ) d x = 2 1 tan 2 ( x ) + ln ∣ cos ( x ) ∣ + C ∫ cot 3 ( x ) d x = − 1 2 cot 2 ( x ) − ln ∣ sin ( x ) ∣ + C \int{{\cot}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{2}}{{\cot}}^{{2}}{\left({x}\right)}-{\ln}{\left|{\sin{{\left({x}\right)}}}\right|}+{C} ∫ cot 3 ( x ) d x = − 2 1 cot 2 ( x ) − ln ∣ sin ( x ) ∣ + C ∫ sec 3 ( x ) d x = 1 2 sec ( x ) tan ( x ) + 1 2 ln ∣ sec ( x ) + tan ( x ) ∣ + C \int{{\sec}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}+\frac{{1}}{{2}}{\ln}{\left|{\sec{{\left({x}\right)}}}+{\tan{{\left({x}\right)}}}\right|}+{C} ∫ sec 3 ( x ) d x = 2 1 sec ( x ) tan ( x ) + 2 1 ln ∣ sec ( x ) + tan ( x ) ∣ + C ∫ csc 3 ( x ) d x = − 1 2 csc ( x ) cot ( x ) + 1 2 ln ∣ csc ( x ) − cot ( x ) ∣ + C \int{{\csc}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{2}}{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}+\frac{{1}}{{2}}{\ln}{\left|{\csc{{\left({x}\right)}}}-{\cot{{\left({x}\right)}}}\right|}+{C} ∫ csc 3 ( x ) d x = − 2 1 csc ( x ) cot ( x ) + 2 1 ln ∣ csc ( x ) − cot ( x ) ∣ + C ∫ sin n ( x ) d x = − 1 n sin n − 1 ( x ) cos ( x ) + n − 1 n ∫ sin n − 2 ( x ) d x \int{{\sin}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{n}}{{\sin}}^{{{n}-{1}}}{\left({x}\right)}{\cos{{\left({x}\right)}}}+\frac{{{n}-{1}}}{{n}}\int{{\sin}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ sin n ( x ) d x = − n 1 sin n − 1 ( x ) cos ( x ) + n n − 1 ∫ sin n − 2 ( x ) d x ∫ cos n ( x ) d x = 1 n cos n − 1 ( x ) sin ( x ) + n − 1 n ∫ cos n − 2 ( x ) d x \int{{\cos}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{n}}{{\cos}}^{{{n}-{1}}}{\left({x}\right)}{\sin{{\left({x}\right)}}}+\frac{{{n}-{1}}}{{n}}\int{{\cos}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ cos n ( x ) d x = n 1 cos n − 1 ( x ) sin ( x ) + n n − 1 ∫ cos n − 2 ( x ) d x ∫ tan n ( x ) d x = 1 n − 1 tan n − 1 ( x ) − ∫ tan n − 2 ( x ) d x \int{{\tan}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{{n}-{1}}}{{\tan}}^{{{n}-{1}}}{\left({x}\right)}-\int{{\tan}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ tan n ( x ) d x = n − 1 1 tan n − 1 ( x ) − ∫ tan n − 2 ( x ) d x ∫ cot n ( x ) d x = − 1 n − 1 cot n − 1 ( x ) − ∫ cot n − 2 ( x ) d x \int{{\cot}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{{n}-{1}}}{{\cot}}^{{{n}-{1}}}{\left({x}\right)}-\int{{\cot}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ cot n ( x ) d x = − n − 1 1 cot n − 1 ( x ) − ∫ cot n − 2 ( x ) d x ∫ sec n ( x ) d x = 1 n − 1 tan ( x ) sec n − 2 ( x ) + n − 2 n − 1 ∫ sec n − 2 ( x ) d x \int{{\sec}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{{n}-{1}}}{\tan{{\left({x}\right)}}}{{\sec}}^{{{n}-{2}}}{\left({x}\right)}+\frac{{{n}-{2}}}{{{n}-{1}}}\int{{\sec}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ sec n ( x ) d x = n − 1 1 tan ( x ) sec n − 2 ( x ) + n − 1 n − 2 ∫ sec n − 2 ( x ) d x ∫ csc n ( x ) d x = − 1 n − 1 cot ( x ) csc n − 2 ( x ) + n − 2 n − 1 ∫ csc n − 2 ( x ) d x \int{{\csc}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{{n}-{1}}}{\cot{{\left({x}\right)}}}{{\csc}}^{{{n}-{2}}}{\left({x}\right)}+\frac{{{n}-{2}}}{{{n}-{1}}}\int{{\csc}}^{{{n}-{2}}}{\left({x}\right)}{d}{x} ∫ csc n ( x ) d x = − n − 1 1 cot ( x ) csc n − 2 ( x ) + n − 1 n − 2 ∫ csc n − 2 ( x ) d x ∫ sin ( a x ) sin ( b x ) d x = sin ( ( a − b ) x ) 2 ( a − b ) − sin ( ( a + b ) x ) 2 ( a + b ) + C \int{\sin{{\left({a}{x}\right)}}}{\sin{{\left({b}{x}\right)}}}{d}{x}=\frac{{{\sin{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}-\frac{{{\sin{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C} ∫ sin ( a x ) sin ( b x ) d x = 2 ( a − b ) s i n ( ( a − b ) x ) − 2 ( a + b ) s i n ( ( a + b ) x ) + C ∫ cos ( a x ) cos ( b x ) d x = sin ( ( a − b ) x ) 2 ( a − b ) + sin ( ( a + b ) x ) 2 ( a + b ) + C \int{\cos{{\left({a}{x}\right)}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=\frac{{{\sin{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}+\frac{{{\sin{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C} ∫ cos ( a x ) cos ( b x ) d x = 2 ( a − b ) s i n ( ( a − b ) x ) + 2 ( a + b ) s i n ( ( a + b ) x ) + C ∫ sin ( a x ) cos ( b x ) d x = − cos ( ( a − b ) x ) 2 ( a − b ) − cos ( ( a + b ) x ) 2 ( a + b ) + C \int{\sin{{\left({a}{x}\right)}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=-\frac{{{\cos{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}-\frac{{{\cos{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C} ∫ sin ( a x ) cos ( b x ) d x = − 2 ( a − b ) c o s ( ( a − b ) x ) − 2 ( a + b ) c o s ( ( a + b ) x ) + C ∫ x sin ( x ) d x = sin ( x ) − x cos ( x ) + C \int{x}{\sin{{\left({x}\right)}}}{d}{x}={\sin{{\left({x}\right)}}}-{x}{\cos{{\left({x}\right)}}}+{C} ∫ x sin ( x ) d x = sin ( x ) − x cos ( x ) + C ∫ x cos ( x ) d x = cos ( x ) + x sin ( x ) + C \int{x}{\cos{{\left({x}\right)}}}{d}{x}={\cos{{\left({x}\right)}}}+{x}{\sin{{\left({x}\right)}}}+{C} ∫ x cos ( x ) d x = cos ( x ) + x sin ( x ) + C ∫ x n sin ( x ) d x = − x n cos ( x ) + n ∫ x n − 1 cos ( x ) d x \int{{x}}^{{n}}{\sin{{\left({x}\right)}}}{d}{x}=-{{x}}^{{n}}{\cos{{\left({x}\right)}}}+{n}\int{{x}}^{{{n}-{1}}}{\cos{{\left({x}\right)}}}{d}{x} ∫ x n sin ( x ) d x = − x n cos ( x ) + n ∫ x n − 1 cos ( x ) d x ∫ x n cos ( x ) d x = x n sin ( x ) − n ∫ x n − 1 sin ( x ) d x \int{{x}}^{{n}}{\cos{{\left({x}\right)}}}{d}{x}={{x}}^{{n}}{\sin{{\left({x}\right)}}}-{n}\int{{x}}^{{{n}-{1}}}{\sin{{\left({x}\right)}}}{d}{x} ∫ x n cos ( x ) d x = x n sin ( x ) − n ∫ x n − 1 sin ( x ) d x ∫ sin n ( x ) cos m ( x ) d x = − sin n − 1 ( x ) cos m + 1 ( x ) n + m + n − 1 n + m ∫ sin n − 2 ( x ) cos m ( x ) d x = \int{{\sin}}^{{n}}{\left({x}\right)}{{\cos}}^{{m}}{\left({x}\right)}{d}{x}=-\frac{{{{\sin}}^{{{n}-{1}}}{\left({x}\right)}{{\cos}}^{{{m}+{1}}}{\left({x}\right)}}}{{{n}+{m}}}+\frac{{{n}-{1}}}{{{n}+{m}}}\int{{\sin}}^{{{n}-{2}}}{\left({x}\right)}{{\cos}}^{{m}}{\left({x}\right)}{d}{x}= ∫ sin n ( x ) cos m ( x ) d x = − n + m s i n n − 1 ( x ) c o s m + 1 ( x ) + n + m n − 1 ∫ sin n − 2 ( x ) cos m ( x ) d x =
= sin n + 1 ( x ) cos m − 1 ( x ) n + m + m − 1 n + m ∫ sin n ( x ) cos m − 2 ( x ) d x =\frac{{{{\sin}}^{{{n}+{1}}}{\left({x}\right)}{{\cos}}^{{{m}-{1}}}{\left({x}\right)}}}{{{n}+{m}}}+\frac{{{m}-{1}}}{{{n}+{m}}}\int{{\sin}}^{{n}}{\left({x}\right)}{{\cos}}^{{{m}-{2}}}{\left({x}\right)}{d}{x} = n + m s i n n + 1 ( x ) c o s m − 1 ( x ) + n + m m − 1 ∫ sin n ( x ) cos m − 2 ( x ) d x
Inverse Trigonometric Forms ∫ arcsin ( x ) d x = x arcsin ( x ) + 1 − x 2 + C \int{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arcsin}{{\left({x}\right)}}}+\sqrt{{{1}-{{x}}^{{2}}}}+{C} ∫ arcsin ( x ) d x = x arcsin ( x ) + 1 − x 2 + C ∫ arccos ( x ) d x = x arccos ( x ) − 1 − x 2 + C \int{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arccos}{{\left({x}\right)}}}-\sqrt{{{1}-{{x}}^{{2}}}}+{C} ∫ arccos ( x ) d x = x arccos ( x ) − 1 − x 2 + C ∫ arctan ( x ) d x = x arctan ( x ) − 1 2 ln ( 1 + x 2 ) + C \int{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arctan}{{\left({x}\right)}}}-\frac{{1}}{{2}}{\ln{{\left({1}+{{x}}^{{2}}\right)}}}+{C} ∫ arctan ( x ) d x = x arctan ( x ) − 2 1 ln ( 1 + x 2 ) + C ∫ x arcsin ( x ) d x = 2 x 2 − 1 4 arcsin ( x ) + x 1 − x 2 4 + C \int{x}{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{1}}}{{4}}{\operatorname{arcsin}{{\left({x}\right)}}}+\frac{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}{{4}}+{C} ∫ x arcsin ( x ) d x = 4 2 x 2 − 1 arcsin ( x ) + 4 x 1 − x 2 + C ∫ x arccos ( x ) d x = 2 x 2 − 1 4 arccos ( x ) − x 1 − x 2 4 + C \int{x}{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{1}}}{{4}}{\operatorname{arccos}{{\left({x}\right)}}}-\frac{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}{{4}}+{C} ∫ x arccos ( x ) d x = 4 2 x 2 − 1 arccos ( x ) − 4 x 1 − x 2 + C ∫ x arctan ( x ) d x = x 2 + 1 2 arctan ( x ) − x 2 + C \int{x}{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}=\frac{{{{x}}^{{2}}+{1}}}{{2}}{\operatorname{arctan}{{\left({x}\right)}}}-\frac{{x}}{{2}}+{C} ∫ x arctan ( x ) d x = 2 x 2 + 1 arctan ( x ) − 2 x + C ∫ x n arcsin ( x ) d x = 1 n + 1 ( x n + 1 arcsin ( x ) − ∫ x n + 1 1 − x 2 d x ) \int{{x}}^{{n}}{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arcsin}{{\left({x}\right)}}}-\int\frac{{{{x}}^{{{n}+{1}}}}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}{d}{x}\right)} ∫ x n arcsin ( x ) d x = n + 1 1 ( x n + 1 arcsin ( x ) − ∫ 1 − x 2 x n + 1 d x ) n ≠ − 1 {n}\ne-{1} n = − 1 ∫ x n arccos ( x ) d x = 1 n + 1 ( x n + 1 arccos ( x ) + ∫ x n + 1 1 − x 2 d x ) \int{{x}}^{{n}}{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arccos}{{\left({x}\right)}}}+\int\frac{{{{x}}^{{{n}+{1}}}}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}{d}{x}\right)} ∫ x n arccos ( x ) d x = n + 1 1 ( x n + 1 arccos ( x ) + ∫ 1 − x 2 x n + 1 d x ) n ≠ − 1 {n}\ne-{1} n = − 1 ∫ x n arctan ( x ) d x = 1 n + 1 ( x n + 1 arctan ( x ) − ∫ x n + 1 1 + x 2 d x ) \int{{x}}^{{n}}{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arctan}{{\left({x}\right)}}}-\int\frac{{{{x}}^{{{n}+{1}}}}}{{{1}+{{x}}^{{2}}}}{d}{x}\right)} ∫ x n arctan ( x ) d x = n + 1 1 ( x n + 1 arctan ( x ) − ∫ 1 + x 2 x n + 1 d x ) n ≠ − 1 {n}\ne-{1} n = − 1 Hyperbolic Forms ∫ sinh ( x ) d x = cosh ( x ) + C \int{\sinh{{\left({x}\right)}}}{d}{x}={\cosh{{\left({x}\right)}}}+{C} ∫ sinh ( x ) d x = cosh ( x ) + C ∫ cosh ( x ) d x = sinh ( x ) + C \int{\cosh{{\left({x}\right)}}}{d}{x}={\sinh{{\left({x}\right)}}}+{C} ∫ cosh ( x ) d x = sinh ( x ) + C ∫ tanh ( x ) d x = ln ( cosh ( x ) ) + C \int{\tanh{{\left({x}\right)}}}{d}{x}={\ln{{\left({\cosh{{\left({x}\right)}}}\right)}}}+{C} ∫ tanh ( x ) d x = ln ( cosh ( x ) ) + C ∫ coth ( x ) d x = ln ∣ sinh ( x ) ∣ + C \int{\coth{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sinh{{\left({x}\right)}}}\right|}+{C} ∫ coth ( x ) d x = ln ∣ sinh ( x ) ∣ + C ∫ sech ( x ) d x = arctan ∣ sinh ( x ) ∣ + C \int{\operatorname{sech}{{\left({x}\right)}}}{d}{x}={\operatorname{arctan}}{\left|{\sinh{{\left({x}\right)}}}\right|}+{C} ∫ sech ( x ) d x = arctan ∣ sinh ( x ) ∣ + C ∫ csch ( x ) d x = ln ∣ tanh ( 1 2 x ) ∣ + C \int{\operatorname{csch}{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\tanh{{\left(\frac{{1}}{{2}}{x}\right)}}}\right|}+{C} ∫ csch ( x ) d x = ln ∣ ∣ tanh ( 2 1 x ) ∣ ∣ + C ∫ sech 2 ( x ) d x = tanh ( x ) + C \int{{\operatorname{sech}}}^{{2}}{\left({x}\right)}{d}{x}={\tanh{{\left({x}\right)}}}+{C} ∫ sech 2 ( x ) d x = tanh ( x ) + C ∫ csch 2 ( x ) d x = − coth ( x ) + C \int{{\operatorname{csch}}}^{{2}}{\left({x}\right)}{d}{x}=-{\coth{{\left({x}\right)}}}+{C} ∫ csch 2 ( x ) d x = − coth ( x ) + C ∫ sech ( x ) tanh ( x ) d x = − s e c h ( x ) + C \int{\operatorname{sech}{{\left({x}\right)}}}{\tanh{{\left({x}\right)}}}{d}{x}=-{s}{e}{c}{h}{\left({x}\right)}+{C} ∫ sech ( x ) tanh ( x ) d x = − s e c h ( x ) + C ∫ csch ( x ) coth ( x ) d x = − c s c h ( x ) + C \int{\operatorname{csch}{{\left({x}\right)}}}{\coth{{\left({x}\right)}}}{d}{x}=-{c}{s}{c}{h}{\left({x}\right)}+{C} ∫ csch ( x ) coth ( x ) d x = − c s c h ( x ) + C Forms Involving a 2 + x 2 \sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}} a 2 + x 2 a > 0 {a}>{0} a > 0 ∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 ln ( x + a 2 + x 2 ) + C \int\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C} ∫ a 2 + x 2 d x = 2 x a 2 + x 2 + 2 a 2 ln ( x + a 2 + x 2 ) + C ∫ x 2 a 2 + x 2 d x = x 8 ( a 2 + 2 x 2 ) a 2 + x 2 − a 4 8 ln ( x + a 2 + x 2 ) + C \int{{x}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({{a}}^{{2}}+{2}{{x}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-\frac{{{{a}}^{{4}}}}{{8}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C} ∫ x 2 a 2 + x 2 d x = 8 x ( a 2 + 2 x 2 ) a 2 + x 2 − 8 a 4 ln ( x + a 2 + x 2 ) + C ∫ a 2 + x 2 x d x = a 2 + x 2 − a ln ∣ a + a 2 + x 2 x ∣ + C \int\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-{a}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}\right|}+{C} ∫ x a 2 + x 2 d x = a 2 + x 2 − a ln ∣ ∣ x a + a 2 + x 2 ∣ ∣ + C ∫ a 2 + x 2 x 2 d x = − a 2 + x 2 x + ln ( x + a 2 + x 2 ) + C \int\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}+{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C} ∫ x 2 a 2 + x 2 d x = − x a 2 + x 2 + ln ( x + a 2 + x 2 ) + C ∫ d x a 2 + x 2 = ln ( x + a 2 + x 2 ) + C \int\frac{{{d}{x}}}{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}={\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C} ∫ a 2 + x 2 d x = ln ( x + a 2 + x 2 ) + C ∫ x 2 d x a 2 + x 2 = x 2 a 2 + x 2 − a 2 2 ln ( x + a 2 + x 2 ) + C \int\frac{{{{x}}^{{2}}{d}{x}}}{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-\frac{{{{a}}^{{2}}}}{{2}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C} ∫ a 2 + x 2 x 2 d x = 2 x a 2 + x 2 − 2 a 2 ln ( x + a 2 + x 2 ) + C ∫ d x x a 2 + x 2 = − 1 a ln ∣ a 2 + x 2 + a x ∣ + C \int\frac{{{d}{x}}}{{{x}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=-\frac{{1}}{{a}}{\ln}{\left|\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}+{a}}}{{x}}\right|}+{C} ∫ x a 2 + x 2 d x = − a 1 ln ∣ ∣ x a 2 + x 2 + a ∣ ∣ + C ∫ d x x 2 a 2 + x 2 = − a 2 + x 2 a 2 x + C \int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=-\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{{{a}}^{{2}}{x}}}+{C} ∫ x 2 a 2 + x 2 d x = − a 2 x a 2 + x 2 + C ∫ d x ( a 2 + x 2 ) 3 2 = x a 2 a 2 + x 2 + C \int\frac{{{d}{x}}}{{{{\left({{a}}^{{2}}+{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}+{C} ∫ ( a 2 + x 2 ) 2 3 d x = a 2 a 2 + x 2 x + C Forms Involving a 2 − x 2 \sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}} a 2 − x 2 a > 0 {a}>{0} a > 0 ∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 arcsin ( x a ) + C \int\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ a 2 − x 2 d x = 2 x a 2 − x 2 + 2 a 2 arcsin ( a x ) + C ∫ x 2 a 2 − x 2 d x = x 8 ( 2 x 2 − a 2 ) a 2 − x 2 + a 4 8 arcsin ( x a ) + C \int{{x}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{{a}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{4}}}}{{8}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ x 2 a 2 − x 2 d x = 8 x ( 2 x 2 − a 2 ) a 2 − x 2 + 8 a 4 arcsin ( a x ) + C ∫ a 2 − x 2 x = a 2 − x 2 − a ln ∣ a + a 2 − x 2 x ∣ + C \int\frac{{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}=\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}-{a}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}\right|}+{C} ∫ x a 2 − x 2 = a 2 − x 2 − a ln ∣ ∣ x a + a 2 − x 2 ∣ ∣ + C ∫ a 2 − x 2 x 2 d x = − 1 x a 2 − x 2 − arcsin ( x a ) + C \int\frac{{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{1}}{{x}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}-{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ x 2 a 2 − x 2 d x = − x 1 a 2 − x 2 − arcsin ( a x ) + C ∫ x 2 a 2 − x 2 d x = − x 2 a 2 − x 2 + a 2 2 arcsin ( x a ) + C \int\frac{{{{x}}^{{2}}}}{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}{d}{x}=-\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ a 2 − x 2 x 2 d x = − 2 x a 2 − x 2 + 2 a 2 arcsin ( a x ) + C ∫ d x x a 2 − x 2 = − 1 a ln ∣ a + a 2 − x 2 x ∣ + C \int\frac{{{d}{x}}}{{{x}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}=-\frac{{1}}{{a}}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}\right|}+{C} ∫ x a 2 − x 2 d x = − a 1 ln ∣ ∣ x a + a 2 − x 2 ∣ ∣ + C ∫ d x x 2 a 2 − x 2 = − 1 a 2 x a 2 − x 2 + C \int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}=-\frac{{1}}{{{{a}}^{{2}}{x}}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+{C} ∫ x 2 a 2 − x 2 d x = − a 2 x 1 a 2 − x 2 + C ∫ ( a 2 − x 2 ) 3 2 d x = − x 8 ( 2 x 2 − 5 a 2 ) a 2 − x 2 + 3 a 4 8 arcsin ( x a ) + C \int{{\left({{a}}^{{2}}-{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}{d}{x}=-\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{5}{{a}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{3}{{a}}^{{4}}}}{{8}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C} ∫ ( a 2 − x 2 ) 2 3 d x = − 8 x ( 2 x 2 − 5 a 2 ) a 2 − x 2 + 8 3 a 4 arcsin ( a x ) + C ∫ d x ( a 2 − x 2 ) 3 2 = x a 2 a 2 − x 2 + C \int\frac{{{d}{x}}}{{{{\left({{a}}^{{2}}-{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}+{C} ∫ ( a 2 − x 2 ) 2 3 d x = a 2 a 2 − x 2 x + C Forms Involving x 2 − a 2 \sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}} x 2 − a 2 a > 0 {a}>{0} a > 0 ∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 ln ∣ x + x 2 − a 2 ∣ + C \int\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-\frac{{{{a}}^{{2}}}}{{2}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C} ∫ x 2 − a 2 d x = 2 x x 2 − a 2 − 2 a 2 ln ∣ ∣ x + x 2 − a 2 ∣ ∣ + C ∫ x 2 x 2 − a 2 d x = x 8 ( 2 x 2 − a 2 ) x 2 − a 2 − a 4 8 ln ∣ x + x 2 − a 2 ∣ + C \int{{x}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{{a}}^{{2}}\right)}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-\frac{{{{a}}^{{4}}}}{{8}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C} ∫ x 2 x 2 − a 2 d x = 8 x ( 2 x 2 − a 2 ) x 2 − a 2 − 8 a 4 ln ∣ ∣ x + x 2 − a 2 ∣ ∣ + C ∫ x 2 − a 2 x d x = x 2 − a 2 − a ⋅ arccos ( a ∣ u ∣ ) + C \int\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-{a}\cdot{\operatorname{arccos}{{\left(\frac{{a}}{{{\left|{u}\right|}}}\right)}}}+{C} ∫ x x 2 − a 2 d x = x 2 − a 2 − a ⋅ arccos ( ∣ u ∣ a ) + C ∫ x 2 − a 2 x 2 d x = − x 2 − a 2 x + ln ∣ x + x 2 − a 2 ∣ + C \int\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{x}}+{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C} ∫ x 2 x 2 − a 2 d x = − x x 2 − a 2 + ln ∣ ∣ x + x 2 − a 2 ∣ ∣ + C ∫ d x x 2 − a 2 = ln ∣ x + x 2 − a 2 ∣ + C \int\frac{{{d}{x}}}{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}={\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C} ∫ x 2 − a 2 d x = ln ∣ ∣ x + x 2 − a 2 ∣ ∣ + C ∫ x 2 x 2 − a 2 d x = x 2 x 2 − a 2 + a 2 2 ln ∣ x + x 2 − a 2 ∣ + C \int\frac{{{{x}}^{{2}}}}{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C} ∫ x 2 − a 2 x 2 d x = 2 x x 2 − a 2 + 2 a 2 ln ∣ ∣ x + x 2 − a 2 ∣ ∣ + C ∫ d x x 2 x 2 − a 2 = x 2 − a 2 a 2 x + C \int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}=\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{{{a}}^{{2}}{x}}}+{C} ∫ x 2 x 2 − a 2 d x = a 2 x x 2 − a 2 + C ∫ d x ( x 2 − a 2 ) 3 2 = − x a 2 x 2 − a 2 + C \int\frac{{{d}{x}}}{{{{\left({{x}}^{{2}}-{{a}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=-\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}+{C} ∫ ( x 2 − a 2 ) 2 3 d x = − a 2 x 2 − a 2 x + C Forms Involving a + b x {a}+{b}{x} a + b x ∫ x a + b x d x = 1 b 2 ( a + b x − a ln ∣ a + b x ∣ ) + C \int\frac{{x}}{{{a}+{b}{x}}}{d}{x}=\frac{{1}}{{{{b}}^{{2}}}}{\left({a}+{b}{x}-{a}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C} ∫ a + b x x d x = b 2 1 ( a + b x − a ln ∣ a + b x ∣ ) + C ∫ x 2 a + b x d x = 1 2 b 3 ( ( a + b x ) 2 − 4 a ( a + b x ) + 2 a 2 ln ∣ a + b x ∣ ) + C \int\frac{{{{x}}^{{2}}}}{{{a}+{b}{x}}}{d}{x}=\frac{{1}}{{{2}{{b}}^{{3}}}}{\left({{\left({a}+{b}{x}\right)}}^{{2}}-{4}{a}{\left({a}+{b}{x}\right)}+{2}{{a}}^{{2}}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C} ∫ a + b x x 2 d x = 2 b 3 1 ( ( a + b x ) 2 − 4 a ( a + b x ) + 2 a 2 ln ∣ a + b x ∣ ) + C ∫ d x x ( a + b x ) = 1 a ln ∣ x a + b x ∣ + C \int\frac{{{d}{x}}}{{{x}{\left({a}+{b}{x}\right)}}}=\frac{{1}}{{a}}{\ln}{\left|\frac{{x}}{{{a}+{b}{x}}}\right|}+{C} ∫ x ( a + b x ) d x = a 1 ln ∣ ∣ a + b x x ∣ ∣ + C ∫ d x x 2 ( a + b x ) = − 1 a x + b a 2 ln ∣ a + b x x ∣ + C \int\frac{{{d}{x}}}{{{{x}}^{{2}}{\left({a}+{b}{x}\right)}}}=-\frac{{1}}{{{a}{x}}}+\frac{{b}}{{{{a}}^{{2}}}}{\ln}{\left|\frac{{{a}+{b}{x}}}{{x}}\right|}+{C} ∫ x 2 ( a + b x ) d x = − a x 1 + a 2 b ln ∣ ∣ x a + b x ∣ ∣ + C ∫ x ( a + b x ) 2 d x = a b 2 ( a + b x ) + 1 b 2 ln ∣ a + b x ∣ + C \int\frac{{x}}{{{{\left({a}+{b}{x}\right)}}^{{2}}}}{d}{x}=\frac{{a}}{{{{b}}^{{2}}{\left({a}+{b}{x}\right)}}}+\frac{{1}}{{{{b}}^{{2}}}}{\ln}{\left|{a}+{b}{x}\right|}+{C} ∫ ( a + b x ) 2 x d x = b 2 ( a + b x ) a + b 2 1 ln ∣ a + b x ∣ + C ∫ d x x ( a + b x ) 2 = 1 a ( a + b x ) − 1 a 2 ln ∣ a + b x x ∣ + C \int\frac{{{d}{x}}}{{{x}{{\left({a}+{b}{x}\right)}}^{{2}}}}=\frac{{1}}{{{a}{\left({a}+{b}{x}\right)}}}-\frac{{1}}{{{{a}}^{{2}}}}{\ln}{\left|\frac{{{a}+{b}{x}}}{{x}}\right|}+{C} ∫ x ( a + b x ) 2 d x = a ( a + b x ) 1 − a 2 1 ln ∣ ∣ x a + b x ∣ ∣ + C ∫ x 2 ( a + b x ) 2 d x = 1 b 3 ( a + b x − a 2 a + b x − 2 a ln ∣ a + b x ∣ ) + C \int\frac{{{{x}}^{{2}}}}{{{{\left({a}+{b}{x}\right)}}^{{2}}}}{d}{x}=\frac{{1}}{{{{b}}^{{3}}}}{\left({a}+{b}{x}-\frac{{{{a}}^{{2}}}}{{{a}+{b}{x}}}-{2}{a}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C} ∫ ( a + b x ) 2 x 2 d x = b 3 1 ( a + b x − a + b x a 2 − 2 a ln ∣ a + b x ∣ ) + C ∫ x a + b x d x = 2 15 b 2 ( 3 b x − 2 a ) ( a + b x ) 3 2 + C \int{x}\sqrt{{{a}+{b}{x}}}{d}{x}=\frac{{2}}{{{15}{{b}}^{{2}}}}{\left({3}{b}{x}-{2}{a}\right)}{{\left({a}+{b}{x}\right)}}^{{\frac{{3}}{{2}}}}+{C} ∫ x a + b x d x = 15 b 2 2 ( 3 b x − 2 a ) ( a + b x ) 2 3 + C ∫ x a + b x d x = 2 3 b 2 ( b x − 2 a ) a + b x + C \int\frac{{x}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{2}}{{{3}{{b}}^{{2}}}}{\left({b}{x}-{2}{a}\right)}\sqrt{{{a}+{b}{x}}}+{C} ∫ a + b x x d x = 3 b 2 2 ( b x − 2 a ) a + b x + C ∫ x 2 a + b x d x = 2 15 b 3 ( 8 a 2 + 3 b 2 x 2 − 4 a b x ) a + b x + C \int\frac{{{{x}}^{{2}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{2}}{{{15}{{b}}^{{3}}}}{\left({8}{{a}}^{{2}}+{3}{{b}}^{{2}}{{x}}^{{2}}-{4}{a}{b}{x}\right)}\sqrt{{{a}+{b}{x}}}+{C} ∫ a + b x x 2 d x = 15 b 3 2 ( 8 a 2 + 3 b 2 x 2 − 4 a b x ) a + b x + C ∫ d x x a + b x = { 1 a ln ∣ a + b x − a a + b x + a ∣ + C if a > 0 2 − a arctan ( a + b x − a ) + C if a < 0 \int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}}={\left\{\begin{array}{c}\frac{{1}}{{\sqrt{{{a}}}}}{\ln}{\left|\frac{{\sqrt{{{a}+{b}{x}}}-\sqrt{{{a}}}}}{{\sqrt{{{a}+{b}{x}}}+\sqrt{{{a}}}}}\right|}+{C}{\quad\text{if}\quad}{a}>{0}\\\frac{{2}}{{\sqrt{{-{a}}}}}{\operatorname{arctan}{{\left(\sqrt{{\frac{{{a}+{b}{x}}}{{-{a}}}}}\right)}}}+{C}{\quad\text{if}\quad}{a}<{0}\\ \end{array}\right.} ∫ x a + b x d x = ⎩ ⎨ ⎧ a 1 ln ∣ ∣ a + b x + a a + b x − a ∣ ∣ + C if a > 0 − a 2 arctan ( − a a + b x ) + C if a < 0 ∫ a + b x x d x = 2 a + b x + a ∫ d x x a + b x \int\frac{{\sqrt{{{a}+{b}{x}}}}}{{x}}{d}{x}={2}\sqrt{{{a}+{b}{x}}}+{a}\int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}} ∫ x a + b x d x = 2 a + b x + a ∫ x a + b x d x ∫ a + b x x 2 d x = − a + b x x + b 2 ∫ d x x a + b x \int\frac{{\sqrt{{{a}+{b}{x}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{a}+{b}{x}}}}}{{x}}+\frac{{b}}{{2}}\int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}} ∫ x 2 a + b x d x = − x a + b x + 2 b ∫ x a + b x d x ∫ x n a + b x d x = 2 b ( 2 n + 3 ) ( x n ( a + b x ) 3 2 − n a ∫ x n − 1 a + b x d x ) \int{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}{d}{x}=\frac{{2}}{{{b}{\left({2}{n}+{3}\right)}}}{\left({{x}}^{{n}}{{\left({a}+{b}{x}\right)}}^{{\frac{{3}}{{2}}}}-{n}{a}\int{{x}}^{{{n}-{1}}}\sqrt{{{a}+{b}{x}}}{d}{x}\right)} ∫ x n a + b x d x = b ( 2 n + 3 ) 2 ( x n ( a + b x ) 2 3 − n a ∫ x n − 1 a + b x d x ) ∫ x n a + b x d x = 2 x n a + b x b ( 2 n + 1 ) − 2 n a b ( 2 n + 1 ) ∫ x n − 1 a + b x d x \int\frac{{{{x}}^{{n}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{{2}{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}}}{{{b}{\left({2}{n}+{1}\right)}}}-\frac{{{2}{n}{a}}}{{{b}{\left({2}{n}+{1}\right)}}}\int\frac{{{{x}}^{{{n}-{1}}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x} ∫ a + b x x n d x = b ( 2 n + 1 ) 2 x n a + b x − b ( 2 n + 1 ) 2 n a ∫ a + b x x n − 1 d x ∫ d x x n a + b x = − a + b x a ( n − 1 ) x n − 1 − b ( 2 n − 3 ) 2 a ( n − 1 ) ∫ d x x n − 1 a + b x \int\frac{{{d}{x}}}{{{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}}}=-\frac{{\sqrt{{{a}+{b}{x}}}}}{{{a}{\left({n}-{1}\right)}{{x}}^{{{n}-{1}}}}}-\frac{{{b}{\left({2}{n}-{3}\right)}}}{{{2}{a}{\left({n}-{1}\right)}}}\int\frac{{{d}{x}}}{{{{x}}^{{{n}-{1}}}\sqrt{{{a}+{b}{x}}}}} ∫ x n a + b x d x = − a ( n − 1 ) x n − 1 a + b x − 2 a ( n − 1 ) b ( 2 n − 3 ) ∫ x n − 1 a + b x d x Forms Involving 2 a x − x 2 \sqrt{{{2}{a}{x}-{{x}}^{{2}}}} 2 a x − x 2 a > 0 {a}>{0} a > 0 ∫ 2 a x − x 2 d x = x − a 2 2 a x − x 2 + a 2 2 arccos ( a − x a ) + C \int\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}{d}{x}=\frac{{{x}-{a}}}{{2}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ 2 a x − x 2 d x = 2 x − a 2 a x − x 2 + 2 a 2 arccos ( a a − x ) + C ∫ x 2 a x − x 2 d x = 2 x 2 − a x − 3 a 2 6 2 a x − x 2 + a 3 2 arccos ( a − x a ) + C \int{x}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{a}{x}-{3}{{a}}^{{2}}}}{{6}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{{a}}^{{3}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ x 2 a x − x 2 d x = 6 2 x 2 − a x − 3 a 2 2 a x − x 2 + 2 a 3 arccos ( a a − x ) + C ∫ 2 a x − x 2 x d x = 2 a x − x 2 + a arccos ( a − x a ) + C \int\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+{a}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ x 2 a x − x 2 d x = 2 a x − x 2 + a arccos ( a a − x ) + C ∫ 2 a x − x 2 x 2 d x = − 2 2 a x − x 2 x − arccos ( a − x a ) + C \int\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{{2}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{x}}-{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ x 2 2 a x − x 2 d x = − x 2 2 a x − x 2 − arccos ( a a − x ) + C ∫ d x 2 a x − x 2 = arccos ( a − x a ) + C \int\frac{{{d}{x}}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}={\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ 2 a x − x 2 d x = arccos ( a a − x ) + C ∫ x 2 a x − x 2 d x = − 2 a x − x 2 + a arccos ( a − x a ) + C \int\frac{{x}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{d}{x}=-\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+{a}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ 2 a x − x 2 x d x = − 2 a x − x 2 + a arccos ( a a − x ) + C ∫ x 2 2 a x − x 2 d x = − x + 3 a 2 2 a x − x 2 + 3 a 2 2 arccos ( a − x a ) + C \int\frac{{{{x}}^{{2}}}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{d}{x}=-\frac{{{x}+{3}{a}}}{{2}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{3}{{a}}^{{2}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C} ∫ 2 a x − x 2 x 2 d x = − 2 x + 3 a 2 a x − x 2 + 2 3 a 2 arccos ( a a − x ) + C ∫ d x x 2 a x − x 2 = − 2 a x − x 2 a x + C \int\frac{{{d}{x}}}{{{x}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}=-\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{{a}{x}}}+{C} ∫ x 2 a x − x 2 d x = − a x 2 a x − x 2 + C 