1.
adx=ax+C2\int a dx=ax+C2
2.
x2dx=x22+C\int x^2 dx=\dfrac{x^2}{2}+C
3.
x2dx=x33+C\int x^2 dx=\dfrac{x^3}{3}+C
4.
xpdx=xp+1p+1+C,p1\int x^p dx=\dfrac{x^{p+1}}{p+1}+C,p \ne 1
5.
(ax+b)ndx=(ax+b)n+1a(n+1)+C,n1\int \left(ax+b\right)^n \,dx=\dfrac{\left(ax+b\right)^{n+1}}{a(n+1)}+C, n \ne 1
6.
dxx=lnx+C\int \dfrac{dx}{x} = \ln|x|+C
7.
1ax+bdx=1alnax+b+C\int \dfrac{1}{ax+b}dx=\dfrac{1}{a}\ln|ax+b|+C
8.
ax+bcx+ddx=acx+bcadc2lncx+d+C\int \dfrac{ax+b}{cx+d}dx=\dfrac{a}{c}x+\dfrac{bc-ad}{c^2} \ln|cx+d|+C
9.
dx(x+a)(x+b)dx=1ab+lnx+bx+a+C,ab\int \dfrac{dx}{(x+a)(x+b)} dx=\dfrac{1}{a-b}+\ln\left| \dfrac{x+b}{x+a}\right|+C, a\ne b
10.
xa+bxdx=1b2(a+bxalna+bx)+C\int \dfrac{x}{a+bx}dx=\dfrac{1}{b^2}\left(a+bx-a\ln|a+bx|\right) + C
11.
x2dxa+bx=1b3[12(a+bx)22a(a+bx)+a2lna+bx]+C\int \dfrac{x^2 dx}{a+bx}=\dfrac{1}{b^3}\left[ \dfrac{1}{2} (a+bx)^2 - 2a(a+bx)+a^2\ln|a+bx|\right]+C
12.
dxx(a+bx)dx=1alna+bxx+C\int \dfrac{dx}{x(a+bx)}dx=\dfrac{1}{a} \ln \left| \dfrac{a+bx}{x} \right| + C
13.
dxx2(a+bx)dx=1ax+ba2lna+bxx+C\int \dfrac{dx}{x^2(a+bx)}dx=-\dfrac{1}{ax}+\dfrac{b}{a^2} \ln\left|\dfrac{a+bx}{x}\right|+C
14.
xdx(a+bx)2=1b2(lna+bx+aa+bx)+C\int \dfrac{xdx}{\left(a+bx\right)^2}=\dfrac{1}{b^2} \left( \ln \left|a+bx\right|+\dfrac{a}{a+bx}\right) + C
15.
x2(a+bx)2=1b3(a+bx2alna+bxa2a+bx)+C\int \dfrac{x^2}{\left(a+bx\right)^2}=\dfrac{1}{b^3} \left(a+bx-2a\ln|a+bx|-\dfrac{a^2}{a+bx}\right)+C
16.
dxx(a+bx)2=1a(a+bx)+1a2lna+bxx+C\int \dfrac{dx}{x(a+bx)^2}=\dfrac{1}{a(a+bx)}+\dfrac{1}{a^2}\ln\left|\dfrac{a+bx}{x}\right|+C
17.
dxx21=12lnx1x+1+C\int \dfrac{dx}{x^2-1}=\dfrac{1}{2}\ln\left|\dfrac{x-1}{x+1}\right|+C
18.
dx1x2=12ln1+x1x+C\int \dfrac{dx}{1-x^2}=\dfrac{1}{2}\ln\left|\dfrac{1+x}{1-x}\right|+C
19.
dxa2x2=12alna+xax+C\int \dfrac{dx}{a^2-x^2}=\dfrac{1}{2a}\ln\left|\dfrac{a+x}{a-x}\right|+C
20.
dxx2a2=12alnxax+a+C\int \dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right|+C
21.
dx1+x2=tan1x+C\int \dfrac{dx}{1+x^2}=\tan^{-1}x+C
22.
dxa2+x2=1atan1xa+C\int \dfrac{dx}{a^2+x^2}=\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+C
23.
dxa+bx2=1abarctan(xab)+C,ab>0\int \dfrac{dx}{a+bx^2}=\dfrac{1}{\sqrt{ab}}\arctan \left(x\sqrt{\dfrac{a}{b}}\right)+C, ab>0
24.
xdxa+bx2=12blnx2+ab+C\int \dfrac{xdx}{a+bx^2}=\dfrac{1}{2b}\ln\left|x^2+\frac{a}{b}\right|+C
25.
dxx(a+bx2)=12alnx2a+bx2+C\int \dfrac{dx}{x\left(a+bx^2\right)}=\dfrac{1}{2a}\ln\left|\dfrac{x^2}{a+bx^2}\right|+C
26.
dxa2b2x2=12ablna+bxabx+C\int \frac{dx}{a^2-b^2x^2}=\dfrac{1}{2ab}\ln\left|\dfrac{a+bx}{a-bx}\right|+C
27.
dxax2+bx+c=1b24acln2ax+bb24ac2ax+b+b24ac+C,b24ac>0\int \dfrac{dx}{ax^2+bx+c}=\dfrac{1}{\sqrt{b^2-4ac}}\ln\left| \dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}} \right|+C, b^2-4ac>0
28.
dxax2+bx+c=24acb2arctan2ax+b4acb2+C\int \dfrac{dx}{ax^2+bx+c}=\dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}}+C