The different methods of Integration will aim at reducing a given Integral to one of the Fundamental or known Integrals . As a matter of facts , there are two principal processes :

**1) The method of substitution , i.e. the change of the independent variable .**

Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let

*ƒ*and

*ϕ*be two functions satisfying the above hypothesis that

*ƒ*is continuous on

*I*and

*ϕ*′ is continuous on the closed interval [

*a*,

*b*]. Then the function

*ƒ*(

*ϕ*(

*t*))

*ϕ*′(

*t*) is also continuous on [

*a*,

*b*]. Hence the integrals

Since

*ƒ*is continuous, it possesses an antiderivative

*F*. The composite function

*F*∘

*ϕ*is then defined. Since

*F*and

*ϕ*are differentiable, the chain rule gives

**2) Integration by Parts ;**

Integrating the product rule for three multiplied functions,

*u*(

*x*),

*v*(

*x*),

*w*(

*x*), gives a similar result:

*n*factors

It may be noted that classes of integrals which are reducible to one or other of the fundamental forms by the above processes are very limited , and that the large majority of the expressions, under proper restrictions , can only be integrated by the aid of infinite series , and in some cases when the integrand involves expression under a radical sign containing powers of x beyond the second , the investigation of such integrals has necessitated the introduction of higher classes of transcendental function such as elliptic functions etc .