Standared Methods of Integration

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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
\int_{\phi(a)}^{\phi(b)} f(x)\,dx
and
\int_a^b f(\phi(t))\phi'(t)\,dt
in fact exist, and it remains to show that they are equal.
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
(F \circ \phi)'(t) = F'(\phi(t))\phi'(t) = f(\phi(t))\phi'(t).
Applying the fundamental theorem of calculus twice gives
\begin{align} \int_a^b f(\phi(t))\phi'(t)\,dt & {} = (F \circ \phi)(b) - (F \circ \phi)(a) \\ & {} = F(\phi(b)) - F(\phi(a)) \\ & {} = \int_{\phi(a)}^{\phi(b)} f(x)\,dx, \end{align}
which is the substitution rule.

2) Integration by Parts ;


Integrating the product rule for three multiplied functions, u(x), v(x), w(x), gives a similar result:
\int_a^b u v \, dw = u v w - \int_a^b u w \, dv - \int_a^b v w \, du.
In general for n factors
\frac{d}{dx} \left(\prod_{i=1}^n u_i(x) \right)= \sum_{j=1}^n \prod_{i\neq j}^n u_i(x) \frac{du_j(x)}{dx},
which leads to
\Bigl[ \prod_{i=1}^n u_i(x) \Bigr]_a^b = \sum_{j=1}^n \int_a^b \prod_{i\neq j}^n u_i(x) \, du_j(x),
where the product is of all functions except for the one differentiated in the same term.

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 .

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