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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 987_integration.JPG

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.

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 

1124_integration.JPG

B. Trigonometric twins

422_integration.JPG

Method of evaluating these integral are illustrated by mean of the following examples:

Example -9: Evaluate ∫ 5/ 1+ x4 dx.

1921_sol.JPG

For I, we write x + 1/x t = => (1 – 1/x2 )dx = dt

                       1952_integration.JPG

Combining the two integrals, we get

                        2141_integration.JPG

Example -10: Evaluate ∫√tanx dx.

Solution:       Put tanx = t2 => sec2x dx = 2t dt

                        => dx = 2tdt / 1 + t4

588_integration.JPG

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.

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