Formulas

1. Standard Integrals

1. \(\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne -1\)
2. \(\displaystyle \int \frac{1}{x}\,dx=\ln|x|+C\)
3. \(\displaystyle \int (ax+b)^n\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,\qquad n\ne -1\)
4. \(\displaystyle \int e^x\,dx=e^x+C\)
5. \(\displaystyle \int a^x\,dx=\frac{a^x}{\ln a}+C,\qquad a>0,\ a\ne1\)
6. \(\displaystyle \int \sin x\,dx=-\cos x+C\)
7. \(\displaystyle \int \cos x\,dx=\sin x+C\)
8. \(\displaystyle \int \tan x\,dx=-\ln|\cos x|+C\)
9. \(\displaystyle \int \cot x\,dx=\ln|\sin x|+C\)
10. \(\displaystyle \int \sec^2x\,dx=\tan x+C\)
11. \(\displaystyle \int \csc^2x\,dx=-\cot x+C\)
12. \(\displaystyle \int \sec x\tan x\,dx=\sec x+C\)
13. \(\displaystyle \int \csc x\cot x\,dx=-\csc x+C\)
14. \(\displaystyle \int \sec x\,dx=\ln|\sec x+\tan x|+C\)
15. \(\displaystyle \int \csc x\,dx=\ln|\csc x-\cot x|+C\)
16. \(\displaystyle \int \sin(ax)\,dx=-\frac1a\cos(ax)+C\)
17. \(\displaystyle \int \cos(ax)\,dx=\frac1a\sin(ax)+C\)
18. \(\displaystyle \int \tan(ax)\,dx=-\frac1a\ln|\cos(ax)|+C\)
19. \(\displaystyle \int \cot(ax)\,dx=\frac1a\ln|\sin(ax)|+C\)
20. \(\displaystyle \int e^{ax}\,dx=\frac1a e^{ax}+C,\qquad a\ne0\)
21. \(\displaystyle \int a^{ax}\,dx=\frac{a^{ax}}{a\ln a}+C,\qquad a>0,\ a\ne1\)
22. \(\displaystyle \int \sqrt{x}\,dx=\int x^{1/2}\,dx=\frac23x^{3/2}+C\)
23. \(\displaystyle \int \frac1{\sqrt{x}}\,dx=\int x^{-1/2}\,dx=2\sqrt{x}+C\)
24. \(\displaystyle \int \frac1{x^2}\,dx=\int x^{-2}\,dx=-\frac1x+C\)

2. Trigonometric Integrals

1. \(\displaystyle \int\sin^2x\,dx=\frac{x}{2}-\frac{\sin2x}{4}+C\)
2. \(\displaystyle \int\cos^2x\,dx=\frac{x}{2}+\frac{\sin2x}{4}+C\)
3. \(\displaystyle \int\sin x\cos x\,dx=\frac{\sin^2x}{2}+C\)
4. \(\displaystyle \int\frac{dx}{\sin^2x}=\int\csc^2x\,dx=-\cot x+C\)
5. \(\displaystyle \int\frac{dx}{\cos^2x}=\int\sec^2x\,dx=\tan x+C\)
6. \(\displaystyle \int\frac{dx}{1+x^2}=\tan^{-1}x+C\)
7. \(\displaystyle \int\frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+C,\qquad |x|<1\)
8. \(\displaystyle \int\frac{dx}{\sqrt{1+x^2}}=\sinh^{-1}x+C =\ln\left|x+\sqrt{1+x^2}\right|+C\)
9. \(\displaystyle \int\tan^2x\,dx=\int(\sec^2x-1)\,dx=\tan x-x+C\)
10. \(\displaystyle \int\cot^2x\,dx=\int(\csc^2x-1)\,dx=-\cot x-x+C\)
11. \(\displaystyle \int(1+\tan^2x)\,dx=\int\sec^2x\,dx=\tan x+C\)
12. \(\displaystyle \int(1+\cot^2x)\,dx=\int\csc^2x\,dx=-\cot x+C\)
13. \(\displaystyle \int\sec^2x\tan x\,dx=\frac{\sec^2x}{2}+C\)
14. \(\displaystyle \int\csc^2x\cot x\,dx=-\frac{\csc^2x}{2}+C\)
15. \(\displaystyle \int\sec x\tan x\,dx=\sec x+C\)
16. \(\displaystyle \int\csc x\cot x\,dx=-\csc x+C\)

3. Inverse Trigonometric and Related Integrals

1. \(\displaystyle \int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\left(\frac{x}{a}\right)+C,\qquad a>0\)
2. \(\displaystyle \int\frac{dx}{a^2+x^2}=\frac1a\tan^{-1}\left(\frac{x}{a}\right)+C,\qquad a>0\)
3. \(\displaystyle \int\frac{dx}{x\sqrt{x^2-a^2}}=\frac1a\sec^{-1}\left(\frac{|x|}{a}\right)+C,\qquad |x|>a>0\)
4. \(\displaystyle \int\frac{dx}{\sqrt{x^2-a^2}}=\ln\left|x+\sqrt{x^2-a^2}\right|+C,\qquad |x|>a>0\)
5. \(\displaystyle \int\frac{dx}{\sqrt{a^2+x^2}}=\ln\left|x+\sqrt{a^2+x^2}\right|+C,\qquad a>0\)
6. \(\displaystyle \int\frac{dx}{ax^2+bx+c} =\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)+C,\qquad 4ac-b^2>0\)
7. \(\displaystyle \int\frac{dx}{ax^2+bx+c} =\frac{1}{\sqrt{b^2-4ac}} \ln\left| \frac{2ax+b-\sqrt{b^2-4ac}} {2ax+b+\sqrt{b^2-4ac}} \right|+C,\qquad b^2-4ac>0\)
8. \(\displaystyle \int\frac{dx}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C,\qquad a\ne0\)
9. \(\displaystyle \int\frac{dx}{(x-a)^2+b^2}=\frac1b\tan^{-1}\left(\frac{x-a}{b}\right)+C,\qquad b>0\)
10. \(\displaystyle \int\frac{dx}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C\)
11. \(\displaystyle \int\frac{x\,dx}{x^2+a^2}=\frac12\ln(x^2+a^2)+C\)
12. \(\displaystyle \int\frac{x\,dx}{x^2-a^2}=\frac12\ln|x^2-a^2|+C\)
13. \(\displaystyle \int\frac{dx}{a^2-x^2}=\frac1{2a}\ln\left|\frac{a+x}{a-x}\right|+C,\qquad |x|
14. \(\displaystyle \int\frac{dx}{x\sqrt{x^2+a^2}} =-\frac1a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|+C\)
15. \(\displaystyle \int\frac{dx}{x\sqrt{a^2-x^2}} =-\frac1a\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right|+C\)
16. \(\displaystyle \int\frac{x\,dx}{\sqrt{x^2+a^2}}=\sqrt{x^2+a^2}+C\)

4. Partial Fractions

1. \(\displaystyle \frac{1}{x-a}=\frac{A}{x-a}\)
2. \(\displaystyle \frac{1}{(x-a)^n} =\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_n}{(x-a)^n}\)
3. \(\displaystyle \frac{1}{(x-a)(x-b)} =\frac{A}{x-a}+\frac{B}{x-b},\qquad a\ne b\)
4. \(\displaystyle \frac{1}{(x-a)(x-b)(x-c)} =\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c},\qquad a,b,c\text{ distinct}\)
5. \(\displaystyle \frac{x}{(x-a)(x-b)} =\frac{A}{x-a}+\frac{B}{x-b}\)
6. \(\displaystyle \frac{ax+b}{(x-a)(x-b)} =\frac{A}{x-a}+\frac{B}{x-b}\)
7. \(\displaystyle \frac{ax+b}{(x-a)(x-b)(x-c)} =\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\)
8. \(\displaystyle \frac{P(x)}{Q(x)},\qquad \deg P<\deg Q,\) is decomposed into suitable partial fractions according to the factors of \(Q(x)\).

5. Important Properties of Definite Integration

1. \(\displaystyle \int_a^a f(x)\,dx=0\)
2. \(\displaystyle \int_a^b k\,dx=k(b-a)\)
3. \(\displaystyle \int_a^b[f(x)+g(x)]\,dx =\int_a^b f(x)\,dx+\int_a^b g(x)\,dx\)
4. \(\displaystyle \int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx\)
5. \(\displaystyle \int_a^b f(x)\,dx=F(x)\Big|_a^b=F(b)-F(a),\qquad F'(x)=f(x)\)
6. If \(f(x)\ge0\) for \(a\le x\le b\), then \(\displaystyle \int_a^b f(x)\,dx\ge0\).
7. If \(f\) is even, then \(\displaystyle \int_{-a}^{a}f(x)\,dx=2\int_0^a f(x)\,dx\).
8. If \(f\) is odd, then \(\displaystyle \int_{-a}^{a}f(x)\,dx=0\).

6. General Properties of Integration

1. \(\displaystyle \int[f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx\)
2. \(\displaystyle \int[f(x)-g(x)]\,dx=\int f(x)\,dx-\int g(x)\,dx\)
3. \(\displaystyle \int kf(x)\,dx=k\int f(x)\,dx\)
4. \(\displaystyle \int f(x)\,dx=F(x)+C\quad\Longrightarrow\quad \frac{d}{dx}[F(x)+C]=f(x)\)
5. \(\displaystyle \int_a^b f(x)\,dx=F(b)-F(a)\)
6. \(\displaystyle \int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\)
7. \(\displaystyle \int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx,\qquad a
8. \(\displaystyle \int_{-a}^{a}f(x)\,dx=2\int_0^a f(x)\,dx\quad\text{if }f\text{ is even}\)
9. \(\displaystyle \int_{-a}^{a}f(x)\,dx=0\quad\text{if }f\text{ is odd}\)

7. Substitution Method

\(\displaystyle I=\int f(x)\,dx,\quad x=\phi(t),\quad dx=\phi'(t)\,dt \quad\Longrightarrow\quad I=\int f(\phi(t))\,\phi'(t)\,dt.\)

1. \(\displaystyle \int f(ax+b)\,dx=\frac1a\int f(u)\,du,\qquad u=ax+b,\quad du=a\,dx\)
2. \(\displaystyle \int(ax+b)^n\,dx= \frac{(ax+b)^{n+1}}{a(n+1)}+C,\qquad n\ne-1\)
3. \(\displaystyle \int f(ax^2+bx+c)(2ax+b)\,dx=\int f(u)\,du,\qquad u=ax^2+bx+c,\quad du=(2ax+b)\,dx\)
4. \(\displaystyle \int\sqrt{ax+b}\,dx=\frac{2}{3a}(ax+b)^{3/2}+C\)
5. \(\displaystyle \int\frac{dx}{\sqrt{ax+b}}=\frac2a\sqrt{ax+b}+C\)
6. \(\displaystyle \int\frac{dx}{(ax+b)^2}=-\frac1{a(ax+b)}+C\)
7. \(\displaystyle \int\frac{dx}{(ax+b)^n} =-\frac1{a(n-1)(ax+b)^{\,n-1}}+C,\qquad n\ne1\)
8. \(\displaystyle \int e^{ax+b}\,dx=\frac1a e^{ax+b}+C\)
9. \(\displaystyle \int\sin(ax+b)\,dx=-\frac1a\cos(ax+b)+C\)
10. \(\displaystyle \int\cos(ax+b)\,dx=\frac1a\sin(ax+b)+C\)

8. Integration by Parts

\(\displaystyle \int u\,v\,dx =u\int v\,dx-\int\left[\frac{du}{dx}\left(\int v\,dx\right)\right]dx.\)

\(\displaystyle \frac{dv}{dx}=v',\qquad \frac{du}{dx}=u', \qquad dv=v'\,dx,\qquad du=u'\,dx.\)

1. \(\displaystyle \int x e^{ax}\,dx =e^{ax}\left(\frac{x}{a}-\frac1{a^2}\right)+C\)
2. \(\displaystyle \int x\sin(ax)\,dx =-\frac{x\cos(ax)}a+\frac{\sin(ax)}{a^2}+C\)
3. \(\displaystyle \int x\cos(ax)\,dx =\frac{x\sin(ax)}a+\frac{\cos(ax)}{a^2}+C\)
4. \(\displaystyle \int e^{ax}\sin(bx)\,dx =\frac{e^{ax}}{a^2+b^2}\left(a\sin bx-b\cos bx\right)+C\)

Note: For indefinite integrals, \(C\) denotes the constant of integration.