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If the integrand is of the form f(x)g(x), where g(x) is a function of the integral of f(x), then put integral of f(x) = t.
Example -8: Evaluate
Solution: Integral of the numerator = x3/2 / 3/2
Put x3/2 = t.
We get I = 2/3 ∫dt/√t2 + a3
= 2/3 In |x3/2 + √x3 + a3| + c.
Some time it is useful to write the integral as a sum of two related integrals which can be evaluated by making suitable substitutions. Examples of such integrals are:
A. Algebraic Twins
B. Trigonometric twins
Method of evaluating these integral are illustrated by mean of the following examples:
Example -9: Evaluate ∫ 5/ 1+ x4 dx.
For I2 , we write x + 1/x t = => (1 – 1/x2 )dx = dt
Combining the two integrals, we get
Example -10: Evaluate ∫√tanx dx.
Solution: Put tanx = t2 => sec2x dx = 2t dt
=> dx = 2tdt / 1 + t4
This can be solved by the method used in example (9).
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