# A few integration by parts problems

• kudoushinichi88
For the second integral you can try:u=\ln(9+x^2) to getdu=\frac{2x}{9+x^2}dx and dv=x dx to getx\ln(9+x^2)-\int \frac{2x^2}{9+x^2}dxFor the third integral you can try:u=x^2 to getdu=2xdx and dv=\tan^{-1}x to getx^2

## Homework Statement

Hello. I am doing some problems on integration by parts and got stuck on the following problems. Any help would be appreciated.

i. $$\int \arcsin x dx$$

ii. $$\int_{0}^{1} x \ln (9+x^2) dx$$

iii. $$\int x^2 \arctan x\, dx$$

## Homework Equations

$$u\,du=uv-v\,du$$

## The Attempt at a Solution

i. I tried u= arcsin x and dv=dx, but ended up with an integral which looks worse than the original question.

ii. I used $u= \ln (9+x^2)\,dx$ but ended up with

$$\int_{0}^{1} \frac{x^3}{9+x^2}\,dx$$
which I am stuck with.

iii. I tried u=arctan x and dv= x^2 dx but ended up with a similar looking integral in question ii, which is $$\int x \ln(1+x^2)\,dx$$

You are getting stuck at a point where a simple substitution would finish it. Look at ii). Try the u-substitution u=9+x^2.

#1: you gave up too soon. Show us what you got.

#2: don't integrate by parts yet. Perform a u-substitution first.

#3: your choice for u and dv are fine, but I don't get that integral that's similar to #2. Again, show us what you got.

For 1) look at the integral:
$$\int 1\cdot\sin^{-1}xdx$$
Use u=arcsin x and dv=1