Integral of 1x2\sqrt{1 - x^{2}}

The calculator will find the integral/antiderivative of 1x2\sqrt{1 - x^{2}}, with steps shown.

Related calculator: Definite and Improper Integral Calculator

Please write without any differentials such as dxdx, dydy etc.
Leave empty for autodetection.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find 1x2dx\int \sqrt{1 - x^{2}}\, dx.

Solution

Let x=sin(u)x=\sin{\left(u \right)}.

Then dx=(sin(u))du=cos(u)dudx=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du (steps can be seen »).

Also, it follows that u=asin(x)u=\operatorname{asin}{\left(x \right)}.

Integrand becomes

1x2=1sin2(u)\sqrt{1 - x^{2}} = \sqrt{1 - \sin^{2}{\left( u \right)}}

Use the identity 1sin2(u)=cos2(u)1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}:

1sin2(u)=cos2(u)\sqrt{1 - \sin^{2}{\left( u \right)}}=\sqrt{\cos^{2}{\left( u \right)}}

Assuming that cos(u)0\cos{\left( u \right)} \ge 0, we obtain the following:

cos2(u)=cos(u)\sqrt{\cos^{2}{\left( u \right)}} = \cos{\left( u \right)}

Thus,

1x2dx=cos2(u)du{\color{red}{\int{\sqrt{1 - x^{2}} d x}}} = {\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}

Rewrite the cosine using the power reducing formula cos2(α)=cos(2α)2+12\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2} with α=u\alpha= u :

cos2(u)du=(cos(2u)2+12)du{\color{red}{\int{\cos^{2}{\left(u \right)} d u}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}

Apply the constant multiple rule cf(u)du=cf(u)du\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du with c=12c=\frac{1}{2} and f(u)=cos(2u)+1f{\left(u \right)} = \cos{\left(2 u \right)} + 1:

(cos(2u)2+12)du=((cos(2u)+1)du2){\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}} = {\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}

Integrate term by term:

(cos(2u)+1)du2=(1du+cos(2u)du)2\frac{{\color{red}{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}}}{2} = \frac{{\color{red}{\left(\int{1 d u} + \int{\cos{\left(2 u \right)} d u}\right)}}}{2}

Apply the constant rule cdu=cu\int c\, du = c u with c=1c=1:

cos(2u)du2+1du2=cos(2u)du2+u2\frac{\int{\cos{\left(2 u \right)} d u}}{2} + \frac{{\color{red}{\int{1 d u}}}}{2} = \frac{\int{\cos{\left(2 u \right)} d u}}{2} + \frac{{\color{red}{u}}}{2}

Let v=2uv=2 u.

Then dv=(2u)du=2dudv=\left(2 u\right)^{\prime }du = 2 du (steps can be seen »), and we have that du=dv2du = \frac{dv}{2}.

The integral can be rewritten as

u2+cos(2u)du2=u2+cos(v)2dv2\frac{u}{2} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{2} = \frac{u}{2} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{2}

Apply the constant multiple rule cf(v)dv=cf(v)dv\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv with c=12c=\frac{1}{2} and f(v)=cos(v)f{\left(v \right)} = \cos{\left(v \right)}:

u2+cos(v)2dv2=u2+(cos(v)dv2)2\frac{u}{2} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{2} = \frac{u}{2} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{2}

The integral of the cosine is cos(v)dv=sin(v)\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}:

u2+cos(v)dv4=u2+sin(v)4\frac{u}{2} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{4} = \frac{u}{2} + \frac{{\color{red}{\sin{\left(v \right)}}}}{4}

Recall that v=2uv=2 u:

u2+sin(v)4=u2+sin((2u))4\frac{u}{2} + \frac{\sin{\left({\color{red}{v}} \right)}}{4} = \frac{u}{2} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{4}

Recall that u=asin(x)u=\operatorname{asin}{\left(x \right)}:

sin(2u)4+u2=sin(2asin(x))4+asin(x)2\frac{\sin{\left(2 {\color{red}{u}} \right)}}{4} + \frac{{\color{red}{u}}}{2} = \frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(x \right)}}} \right)}}{4} + \frac{{\color{red}{\operatorname{asin}{\left(x \right)}}}}{2}

Therefore,

1x2dx=sin(2asin(x))4+asin(x)2\int{\sqrt{1 - x^{2}} d x} = \frac{\sin{\left(2 \operatorname{asin}{\left(x \right)} \right)}}{4} + \frac{\operatorname{asin}{\left(x \right)}}{2}

Using the formulas sin(2asin(α))=2α1α2\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}, sin(2acos(α))=2α1α2\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}, cos(2asin(α))=12α2\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}, cos(2acos(α))=2α21\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1, sinh(2asinh(α))=2αα2+1\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}, sinh(2acosh(α))=2αα1α+1\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}, cosh(2asinh(α))=2α2+1\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1, cosh(2acosh(α))=2α21\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1, simplify the expression:

1x2dx=x1x22+asin(x)2\int{\sqrt{1 - x^{2}} d x} = \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2}

Add the constant of integration:

1x2dx=x1x22+asin(x)2+C\int{\sqrt{1 - x^{2}} d x} = \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2}+C

Answer: 1x2dx=x1x22+asin(x)2+C\int{\sqrt{1 - x^{2}} d x}=\frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2}+C