Having trouble solving $\int x(2x+5)^8dx$.

Question

I am not sure how to solve $\int x(2x+5)^8dx$.

I have tried some different things, but nothing seems to give me the cancelling effect I need to solve the integral.

My Process:

Let $u=2x+5$

Then $du = 2dx$ and $dx = \frac{1}{2}du$

so $\int x u^8 \frac{1}{2} \;du = \frac{1}{2}\int x u^8 \; du$

This is where I get stuck.

The answer should be $\frac{1}{40}(2x+5)^{10}-\frac{5}{36}(2x+5)^{9} + C$ according to my textbook.

How can I solve this problem?

Answer 1

Let $\displaystyle I=\int (2x+5)(2x+5)^8\,dx=\int (2x+5)^9\,dx=\frac{1}{20}(2x+5)^{10}$

And let $J=\displaystyle\int(2x+5)^8 \,dx=\frac{1}{18}(2x+5)^9$

I exclude constants until the end.

Then, $\displaystyle \int x(2x+5)^8\,dx=\frac{1}{2}(I-5J)+C=\frac{1}{40}(2x+5)^{10}-\frac{5}{36}(2x+5)^9+C$.

Answer 2

Since u=2x+5, then $x=\frac{u-5}{2}$ . just put it in our equation $${1 \over 4}\int u^8(u-5)du={1 \over 4}({u^{10} \over 10}-{5u^9 \over 9})=\frac{u^9}{4\cdot90}(9u-50)=\frac{(2x+5)^9}{4\cdot90}(18x+45-50)=\frac{(2x+5)^9\cdot(18x+5)}{360}+c$$

I think it's what you should get