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).

To read more, Buy study materials of Indefinite integral comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.