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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.3
Find the integrals of the functions in Exercises 62 to 83.
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Question : 81 of 271
Marks: +1, -0
cos⁡2x(cos⁡x+sin⁡x)2\frac{\cos 2x}{(\cos x+\sin x)^2}
Solution:  
Let I=∫cos⁡2x(cos⁡x+sin⁡x)2 dxI = \int \frac{\cos 2x}{(\cos x+\sin x)^2}\,dx
=∫cos⁡2x−sin⁡2x(cos⁡x+sin⁡x)2 dx= \int \frac{\cos^2 x - \sin^2 x}{(\cos x+\sin x)^2}\,dx
=∫(cos⁡x−sin⁡x)(cos⁡x+sin⁡x)(cos⁡x+sin⁡x)2 dx= \int \frac{(\cos x - \sin x)(\cos x + \sin x)}{(\cos x + \sin x)^2}\,dx
=∫cos⁡x−sin⁡x(cos⁡x+sin⁡x)2 dx= \int \frac{\cos x - \sin x}{(\cos x+\sin x)^2}\,dx
Put cos⁡x+sin⁡x=t\cos x + \sin x = t ⇒(−sin⁡x+cos⁡x) dx=dt\Rightarrow (-\sin x + \cos x)\,dx = dt
∴I=∫dtt=log⁡∣t∣+C\therefore I = \int \frac{dt}{t} = \log|t| + C =log⁡∣cos⁡x+sin⁡x∣+C= \log|\cos x + \sin x| + C
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