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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 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 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.
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).
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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