Question

methods of integration

Answer 1

Dear student,

The different methods of integration include: Integration by Substitution. Integration by Parts. Integration Using Trigonometric Identities.

Integration By Substitution

Sometimes, we can find the integration by introducing a new independent variable. This method is called Integration using substitution.

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

Substituting x = g(t) in the function ∫f(x), we get;

dx/dt = g'(t)

or dx = g'(t).dt

Thus, I = ∫f(x).dx = f(g(t)).g'(t).dt

Integration By Parts

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;

∫f(x).g(x).dx = f(x).∫g(x).dx–∫(f′(x).∫g(x).dx).dx

Which says:

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

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

ILATE STANDS FOR :- inverse ,logarithmic , arithmatic, trignometry, exponential

Integration Using Trigonometric Identities

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

Few of the trigonometric identities are as follows:

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

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

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

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

Regards