## Thursday, 5 December 2013

### Standared Methods of Integration

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 Fϕ 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
$\frac{d}{dx} \left(\prod_{i=1}^n u_i(x) \right)= \sum_{j=1}^n \prod_{i\neq j}^n u_i(x) \frac{du_j(x)}{dx},$
$\Bigl[ \prod_{i=1}^n u_i(x) \Bigr]_a^b = \sum_{j=1}^n \int_a^b \prod_{i\neq j}^n u_i(x) \, du_j(x),$