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:
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 .