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  • Evaluate the integrals in Exercises 1 to 8 using substitution. sinx/ (1+cos^2x)

Evaluate the integrals in Exercises 1 to 8 using substitution.

    Q5.    \int_0^{\frac{\pi}{2}}\frac{\sin x}{1 + \cos^2 x}dx

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\int_0^{\frac{\pi}{2}}\frac{\sin x}{1 + \cos^2 x}dx =I
let \cos x =t\Rightarrow -\sin x dx = dt
when x=0 then t = 1 and when  x=\pi/2 then t = 0

\\I=\int_{1}^{0}\frac{dt}{1+t^2}\\ =[\tan ^{-1}t]^0_1\\ =\pi/4

 

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manish

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