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

 

See also: integration formulas



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