Question

\int \:\left(x^2-2\right)^3x^3dx

Answer 1

$\int(x^2 - 2)^3x^3dx = \begin{bmatrix} u = x^2 - 2 \\ du = 2xdx \\ dx = \frac{1}{2x}du \end{bmatrix} =$

$= \frac{1}{2} \int u^3(u+2)du = \frac{1}{2} \int (u^4 + 2u^3)du = \\ = \frac{1}{2}(\int u^4du + 2\int u^3 du) = \frac{1}{2}(\frac{u^5}{5} + \frac{u^4}{2}) = \\ =\frac{u^5}{10} + \frac{u^4}{4} = \begin{bmatrix} u = x^2 - 2 \end{bmatrix} = \frac{(x^2 - 2)^5}{10} + \frac{(x^2 - 2)^4}{4} + C$