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Evaluate the integral by using substitution. ∫0π2sinx1+cos2xdx - Mathematics

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Question

Evaluate the integral by using substitution.

`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`

Sum

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Solution

`int_0^(pi/2) (sin x)/(1 + cos^2 x) ` dx

Substituting cos x = t,

⇒ - sin x dx = dt

And x = 0, t = 1, x `= pi/2,` t = 0

Hence, `I = - int_1^0 1/(1 + t^2)` dt

`= - [tan^-1 t]_1^0`

`= - [tan^-1 0 - tan^-1 1]`

`= - [0 - pi/4]`

`= pi/4`

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Evaluation of Definite Integrals by Substitution

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Chapter 7: Integrals - Exercise 7.10 [Page 340]

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NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 7 Integrals
Exercise 7.10 | Q 5 | Page 340

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