# Integration Formula List

## Integration formula list

is on of the most important topic for higher class mathematics and physics. It is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, moments and center of mass, exponential growth and decay , probability,and the volume of a solid, among others.

Here I am going to share a list about Integration Formula. Readers can refer these to solve their problems.

### Topics included:

• Some basic formula for Integration
• Some special formula for Integration
• Integration by parts
• Formula for definite Integration
• PDF formula sheet

## Some Basic Formula of Integration

1. $$\int x^{n} dx =\frac{x^{n+1}}{n+1}+C$$
2. $$\int cosx dx=sinx + C$$
3. $$\int sinx dx=-sinx +C$$
4. $$\int sec^2x=tanx+C$$
5. $$\int cosec^2x dx=-cotx +C$$
6. $$\int secx tanx dx=secx+C$$
7. $$\int cosecx cotx dx=-cosecx+ C$$

## Some Special Formula of Integration

1. $$\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x +C$$
2. $$\int\frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C$$
3. $$\int\frac{dx}{1+x^2}=tan^{-1}x +C$$
4. $$\int\frac{dx}{1+x^2}=-cot^{-1}x +C$$
5. $$\int e^x dx=e^x+C$$
6. $$\int a^xdx=\frac{a^x}{loga}+C$$
7. $$\int\frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C$$
8. $$\int\frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C$$
9. $$\int\frac{1}{x}dx=log|x|+C$$
10. $$\int tanx dx=log |secx|+C$$
11. $$\int cotx dx=log|sinx|+ C$$
12. $$\int secx dx=log|secx+tanx|+ C$$
13. $$\int cosecx dx=log|cosecx-cotx|+C$$
14. $$\int\frac{dx}{x^2-a^2}=\frac{1}{2a}log|\frac{x-a}{x+a}|+C$$
15. $$\int\frac{dx}{a^2-x^2}=\frac{1}{2a}log|\frac{a+x}{a-x}|+C$$
16. $$\int \frac{dx}{x^2+a^2}=\frac{1}{a}tan^{-1}\frac{x}{a}+C$$
17. $$\int\frac{dx}{\sqrt{x^2-a^2}}=log|x+\sqrt{x^2-a^2}|+C$$
18. $$\int\frac{dx}{\sqrt{a^2-x^2}}=sin^{-1}\frac{x}{a}+C$$
19. $$\int\frac{dx}{\sqrt{x^2+a^2}}=log|x+\sqrt{x^2+a^2}|+C$$
20. $$\int \sqrt(x^2-a^2)dx=\frac{x}{2}\sqrt(x^2-a^2)-\frac{a^2}{2}log|x+\sqrt{x^2-a^2}+C$$
21. $$\int\sqrt{x^2+a^2} dx =\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}log|x+\sqrt{x^2+a^2} +C$$
1. $$\int\sqrt{a^2-x^2} dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+C$$

## Integration by Parts

1. $$\int f(x) g(x) dx=f(x)\int g(x)dx-\int{f(x)\int g(x) dx}dx$$
2. $$\int e^x[f(x)+ f’(x)]dx=\int e^x f(x) dx +C$$

## Formula for Definite Integration

1. If $$\int f(x)dx=G(x)$$ so,$$\int_{a}^{b}f(x)dx=G(b)-G(a)$$
2. $$\int_{a}^{b} f(x) dx=\int_{a}^{b}f(t)dt$$
3. $$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$$
4. $$\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}{b}f(x)dx$$
5. $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$
6. $$\int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx$$
7. $$\int_{0}^{2a}f(x)dx=\int_{0}^{a} f(x)dx+\int_{0}^{a}f(2a-x)dx$$
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