Evaluate indefinite integral using susbtitution

Question

I have the following integral to evaluate. I think it should be done by substitution but I get stuck midway when I use $u=x^5$ and $du=5x^4dx$

$$\int x^{14}\sqrt{x^5+2}\,dx$$

Answer 1

Let $u=\sqrt{x^{5}+2}$, then $2 u d x=5 x^{4} d x$ $$ \begin{aligned} I &=\int\left(u^{2}-2\right)^{2} u \cdot \frac{2}{5} u d u \\ &=\frac{2}{5} \int\left(u^{6}-4 u^{4}+4 u^{3}\right) d u \\ &=\frac{2}{5}\left(\frac{u^{7}}{7}-\frac{4 u^{5}}{5}+\frac{4 u^{3}}{3}\right)+C \\ &=\frac{2 u^{3}}{525}\left(15 u^{4}-84 u^{2}+140\right)+C \\ &=\frac{2\left(x^{5}+2\right)^{\frac{3}{2}}}{525}\left[15\left(x^{5}+2\right)^{2}-84\left(x^{5}+2\right)+140\right]+C \\ &=\frac{2\left(x^{5}+2\right)^{\frac{3}{2}}}{525}\left(15 x^{10}-24 x^{5}+32\right)+C . \end{aligned} $$ Wish it helps!