Indefinite Integrals

Anti-derivative

If g(x) is derivative of f(x),then f(x) is said to be an anti-derivative or integral) of g(x).

For example cos x is the derivative of sin x ,so sin x is an anti-derivative(or integral) of cos x symbolically written as     and the symbol    is called as integral sign.

Indefinite integrals of standard functions

Algebra Of Integrals

Methods Of Integration

1.Integration by substitution:

To evaluate integral of the form

we substitute g(x)=t and g'(x)dx=dt.

Example:    

                                        putting sin x=v so that cos xdx=dv

                                

                                

2.Integration by parts:

If u and v are two functions of x,then

where 'u' is the first function and 'v' is the second function(function to be integrated).

We use following preference order for the first function (Exponential(E),Trigonometric(T),Algebraic(A),Logarithmic(L),Inverse(I))-ETALI IN SHORT

Example:   

                                              

Note:  

                    

                    

3.Partial fractions and Integration of rational functions

A function of the form  ,where P and Q are polynomials is called Rational function.

Example:  

A rational function is said to be proper if the degree(highest power) of numerator is less that the degree of denominator otherwise it is called improrer.

Example:  is a proper fraction

whereas  is an improper fraction.

So to apply partial fraction method we must have a proper fraction.

So the above improper fraction can be converted to

(1)Linear and non repeated:

(2)Quadratic and non repeated:

(3)Linear and repeated

(4)Quadratic and repeated