Evaluation of definite integral by substitution

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We know that one of the most important method of evaluation of indefinite integral is method of substitution. While using method of substitution to evaluate definite integrals, following steps are involved.We know that one of the most important method of evaluation of indefinite integral is method of substitution. While using method of substitution to evaluate definite integrals, following steps are involved.

Working rule for Evaluating Definite Integral with Suitable Substitution

Suppose we have to evaluate the integral

(1) Let t = g(x) is the suitable substitution.

Differentiating, we get

dt = g'(x) dx

(2) Now the new variable is t.

The upper limit b and the lower limit a are in terms of x. Change these limits to the new variable g(b) and g(a).

(3) Write and express in terms of t.

(4) Integrate with respect to t.

Find the value of the integral between the new limits g(a) and g(b).

This gives integral of

Example:

1. Evaluate the definite integral .

Solution:

Put t = tan^-1x or x = tan t

when x = 0

tan t = 0

when x = 1, tan t =1

2. Evaluate the definite integral

Solution:

= - (x - 5) if x < 5