Integration by parts allows to integrate some non table functions and the product of functions.
$$d(uv)=udv+vdu$$
$$\Downarrow$$
$$udv=d(uv)-vdu\,\,\,\,\, \Rightarrow\,\,\,\int udv=\int d(uv)-\int vdu$$
1
$$\int x\sin{x}dx$$
2
$$\int x\cos^2{x}dx$$
3
$$\int x\sin^3{x}dx$$
4
$$\int\frac{xdx}{\cos^2{x}}$$
5
$$\int x\cot^2{x}dx$$
6
$$\int \frac{x\sin{x}dx}{cos^2{x}}$$
7
$$\int \ln^2{x}dx$$
8
$$\int x^n \ln{x}dx, n\neq -1$$
9
$$\int\Big(\frac{\ln{x}}{x}\Big)^2dx$$
10
$$\int \sqrt{x}\ln^2{x}dx$$
11
$$\int\frac{\ln{x}}{x^2}dx$$
13
$$\int x^2\ln{(1+x)}dx$$
14
$$\int x\ln{\Big(1+\frac{1}{x}\Big)}dx$$
15
$$\int \ln{|x+\sqrt{1+x^2}|}dx$$
16
$$\int x\ln{\Big|\frac{x+1}{x-1}\Big|}dx$$
17
$$\int\frac{x^2}{(1+x^2)^2}dx$$
18
$$\int\sqrt{2-x^2}dx$$
19
$$\int\sqrt{x^2+3}dx$$
20
$$\int\arctan{x}dx$$
21
$$\int\arccos{x}dx$$
22
$$\int x^2\arctan{x}dx$$
23
$$\int x\arcsin{x}dx$$
24
$$\int x\arctan{x}dx$$
25
$$\int\frac{\arcsin{x}}{x^2}dx$$
26
$$\int\frac{(3+2x^2)}{(1+x^2)}\arctan{x}dx$$
27
$$\int xe^{-x}dx$$
28
$$\int x^2 e^{-2x}dx$$
29
$$\int x^3 e^{-x^2}dx$$
30
$$\int e^{2x}\sin{x}dx$$
31
$$\int\cos^2(\ln{x})dx$$
32
$$\int\sin{x}\ln{(\tan{x})}dx$$
33
$$\int\frac{\ln{\cos{x}}dx}{\cos^2{x}}$$
34
$$\int x\arccos{\frac{1}{x}}dx$$
35
$$\int e^{\alpha x}\sin{\beta x}dx$$
