# Methods of Integration

We have already discussed about the introduction to Integration. Different types of integration includes

1. Integration by Substitution
2. Integration by Parts
3. Integration Using trigonometric Identities
4. Integration of Some particular function
5. Integration by partial fraction

Sometimes, it is really difficult to find the integration of a function, thus we can find the integration by introducing a new independent variable,

The given form of integral function (say $\int f(x)$) can be transformed into another by changing the independent variable x to t,

Substituting $\mathbf{x = g(t)}$ in the function $\mathbf{\int f(x)}$,

$\mathbf{\Rightarrow \frac{\mathrm{d} x}{\mathrm{d} t} = g'(t)}$

or $\mathbf{dx = g'(t).dt}$

Thus, $\mathbf{I = \int f(x).dx = f(g(t)).g'(t).dt}$

Integration by parts requires a special technique for integration of a function, where the integrand function is the multiple of two or more function.

Let us consider an integrand function to be $f(x).g(x)$

Mathematically, integration by parts can be represented as

$\int f(x).g(x). dx = f(x). \int g(x).dx – \int \left ( f'(x). \int g(x).dx \right ).dx$

Which says:

Integral of the product of two function = (First function $\times$ Integral of second function) – Integral of [ (differentiation of first function) $\times$ Integral of second function]

For deciding the first and the second functions, one can follow the ILATE rule for integration.

In an integration of a function, if the integrand involves any kind of trigonometric function, then we use trigonometric identities to simplify the function that can be easily integrated.

Few of the trigonometric identities are as follows:

$\sin^{2}x = \frac{1- \cos 2x}{2}$

$\cos^{2} = \frac{1 + \cos 2x}{2}$

$\sin^{3}x = \frac{3 \sin x – \sin 3x}{4}$

$\cos^{3}x = \frac{3 \cos x + \cos 3x}{4}$

All these identities simplify integrand, that can be easily found out.

Integration of some particular function involves some important formulae of integration that can be applied to make other integration into standard form of integrand. The integration of these standard integrand can be easily found using a direct form of integration method.

We know that a Rational Number can be expressed in the form of p/q, where p and q are integers and $q \neq 0$. In a similar way, a rational function is defined as the ratio of two polynomials which can be expressed in the form of $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$..

There are in general two forms of partial fraction:

1. Proper partial fraction:

When the degree of numerator is less than the degree of denominator, then the fraction is known as proper fraction.

2. Improper partial fraction: When the degree of numerator is greater than the degree of denominator then the fraction is known as improper fraction.Thus the fraction can be simplified into simpler partial fractions, that can be be easily integrated.