Integration Formulas

Using Integration formulas is one of the most basic and most used techniques of differentiation .

Integration formulas are directly derived from the formulas of derivatives , As integral is just an inverse function of derivative we can just inverse the formulas used for finding derivatives to find integral formulas.

For example ,

If $F(x)$ is an integral of $f(x)$ , and $F(x) = \ln x$ then we can easily deduce that $f (x) = \frac{1}{x}$.

As derivative of $F(x) = \ln x$  is $\frac{1}{x}$ .

or, $\frac{d}{dx} \ln x = \frac{1}{x}$ (reference: Derivative of logarithmic function )

and  thus, $\int\frac{1}{x} dx = \ln x$

Similarly we can also work on other derivative formulas and find following Integration formulas.

Integration Formulas:

The main integration formulas used to find integral of functions are:

1. $\int x^n dx = \dfrac{x^{n+1}}{n+1} + c$

2. $\int \cos ax dx = \dfrac{\sin ax}{a} + c$

3. $\int \sin ax dx = - \dfrac{\cos ax}{a} + c$

4. $\int \sec ax \tan ax dx = \dfrac{\sec ax}{a} + c$

5. $\int \sec ^2 ax dx = \dfrac{\tan ax}{a} + c$

6. $\int \csc ax \cot ax dx = - \dfrac{\csc ax}{a} + c$

7. $\int \csc ^2 ax dx = - \dfrac{\cot ax}{a} + c$

8. $\int \dfrac{1}{x} dx = \ln x + c$

9. $\int e ^{ax} dx = \dfrac{e^{ax}}{a} + c$

10. $\int (ax + b)^n dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + c$

11. $\int \dfrac{1}{ax + b} dx = \dfrac{\ln (ax + b)}{a} + c$