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Question
Evaluate the integral by using substitution.
`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`
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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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