Integral of x33x2x^{3} - 3 x^{2}

The calculator will find the integral/antiderivative of x33x2x^{3} - 3 x^{2}, with steps shown.

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Find (x33x2)dx\int \left(x^{3} - 3 x^{2}\right)\, dx.

Solution

Integrate term by term:

(x33x2)dx=(3x2dx+x3dx){\color{red}{\int{\left(x^{3} - 3 x^{2}\right)d x}}} = {\color{red}{\left(- \int{3 x^{2} d x} + \int{x^{3} d x}\right)}}

Apply the power rule xndx=xn+1n+1\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1} (n1)\left(n \neq -1 \right) with n=3n=3:

3x2dx+x3dx=3x2dx+x1+31+3=3x2dx+(x44)- \int{3 x^{2} d x} + {\color{red}{\int{x^{3} d x}}}=- \int{3 x^{2} d x} + {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \int{3 x^{2} d x} + {\color{red}{\left(\frac{x^{4}}{4}\right)}}

Apply the constant multiple rule cf(x)dx=cf(x)dx\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx with c=3c=3 and f(x)=x2f{\left(x \right)} = x^{2}:

x443x2dx=x44(3x2dx)\frac{x^{4}}{4} - {\color{red}{\int{3 x^{2} d x}}} = \frac{x^{4}}{4} - {\color{red}{\left(3 \int{x^{2} d x}\right)}}

Apply the power rule xndx=xn+1n+1\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1} (n1)\left(n \neq -1 \right) with n=2n=2:

x443x2dx=x443x1+21+2=x443(x33)\frac{x^{4}}{4} - 3 {\color{red}{\int{x^{2} d x}}}=\frac{x^{4}}{4} - 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{4}}{4} - 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}

Therefore,

(x33x2)dx=x44x3\int{\left(x^{3} - 3 x^{2}\right)d x} = \frac{x^{4}}{4} - x^{3}

Simplify:

(x33x2)dx=x3(x4)4\int{\left(x^{3} - 3 x^{2}\right)d x} = \frac{x^{3} \left(x - 4\right)}{4}

Add the constant of integration:

(x33x2)dx=x3(x4)4+C\int{\left(x^{3} - 3 x^{2}\right)d x} = \frac{x^{3} \left(x - 4\right)}{4}+C

Answer: (x33x2)dx=x3(x4)4+C\int{\left(x^{3} - 3 x^{2}\right)d x}=\frac{x^{3} \left(x - 4\right)}{4}+C