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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.10
Evaluate the definite integrals in Exercises 198 to 205 using substitution.
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Question : 202 of 271
Marks: +1, -0
∫0π/2sin⁡x1+cos⁡2x dx\int\limits_{0}^{\pi/2} \frac{\sin x}{1+\cos^2 x} \, dx
Solution:  
Let I=∫0π/2sin⁡x1+cos⁡2x dxI=\int\limits_{0}^{\pi/2} \frac{\sin x}{1+\cos^2 x} \, dx
Put cos⁡x=t\cos x = t ⇒−sin⁡x dx=dt\Rightarrow -\sin x \, dx = dt
When x=0,t=cos⁡0=1x = 0, t = \cos 0 = 1, When x=π/2x = \pi/2 , t=cos⁡π2=0t = \cos \frac{\pi}{2} = 0
∴I=∫10−dt1+t2=−[tan⁡−1t]10\therefore I = \int\limits_{1}^{0} \frac{-dt}{1+t^2} = -\left[ \tan^{-1} t \right]_{1}^{0}
=[−tan⁡−10+tan⁡−11]= \left[ -\tan^{-1}0 + \tan^{-1}1 \right]
=[0+π4]=π4= \left[ 0 + \frac{\pi}{4} \right] = \frac{\pi}{4}
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