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The value of $\int \frac{\cos^{3} x + \cos^{5} x}{\sin^{2} x + \sin^{4} x} d x$ is

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a

sin⁡x−6tan−1⁡(sin⁡x)+C

b

sin⁡x−2(sin⁡x)−1+C

c

sin⁡x−2(sin⁡x)−1+5tan−1⁡(sin⁡x)+C

d

none of these

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detailed solution

Correct option is D

Since the integrands satisfies. R(sin⁡x,−cos∣x)=−R(sin⁡x,cos⁡x)Therefore, we substitute sin⁡x=t, so that cos⁡xdx=dt.∫cos3⁡x+cos5⁡xsin2⁡x+sin4⁡xdx=∫cos⁡x1−sin2⁡x+1−sin2⁡x2sin2⁡x+sin4⁡xdx=∫1−t22−t2t2+t4dt=∫1+2t2−61+t2dt=t−2t−6tan−1⁡t+C=sin⁡x−2(sin⁡x)−1−6tan−1⁡(sin⁡x)+C.

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