Indirect Substitution

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 = x^3/2 / 3/2

Put x^3/2 = t.

We get I = 2/3 ∫dt/√t² + a³

= 2/3 In |x3/2 + √x³ + a³| + c.

Derived Substitution

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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