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| 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. |
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| Suppose we have to evaluate the integral |
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| (1) Let t = g(x) is the suitable substitution. |
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| Differentiating, we get |
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| dt = g'(x) dx |
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| (2) Now the new variable is t. |
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| 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). |
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(3) Write  and express  in terms of t. |
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(4) Integrate  with respect to t. |
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| Find the value of the integral between the new limits g(a) and g(b). |
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This gives integral of  |
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| Example: |
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1. Evaluate the definite integral  . |
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| Solution: |
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| Put t = tan-1x or x = tan t |
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 when x = 0 |
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| tan t = 0 |
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when x = 1, tan t =1  |
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2. Evaluate the definite integral  |
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| Solution: |
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| = - (x - 5) if x < 5 |
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