int frac{cos^3 x + cos^5 x}{sin^2 x + sin^4 x | vedclass

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Question

(C) Let $I = \int \frac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x} dx$. $= \int \frac{\cos^3 x (1 + \cos^2 x)}{\sin^2 x (1 + \sin^2 x)} dx$ $= \int \fr

Vedclass · Accepted Answer

$\sin x - 2(\sin x)^{-1} - 6 	an^{-1}(\sin x) + c$

Vedclass · Answer

(C) Let $I = \int \frac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x} dx$. $= \int \frac{\cos^3 x (1 + \cos^2 x)}{\sin^2 x (1 + \sin^2 x)} dx$ $= \int \frac{\cos^2 x (1 + \cos^2 x)}{\sin^2 x (1 + \sin^2 x)} \cos x dx$ $= \int \frac{(1 - \sin^2 x) (1 + 1 - \sin^2 x)}{\sin^2 x (1 + \sin^2 x)} \cos x dx$ $= \int \frac{(1 - \sin^2 x) (2 - \sin^2 x)}{\sin^2 x (1 + \sin^2 x)} \cos x dx$. Let $\sin x = t$,then $\cos x dx = dt$. $I = \int \frac{(1 - t^2) (2 - t^2)}{t^2 (1 + t^2)} dt$ $= \int \frac{t^4 - 3t^2 + 2}{t^2 (1 + t^2)} dt$ $= \int \frac{t^2(t^2 + 1) - 4t^2 + 2}{t^2 (1 + t^2)} dt$ $= \int \left( 1 + \frac{2 - 4t^2}{t^2 (1 + t^2)} ight) dt$ $= \int 1 dt + \int \left( \frac{2}{t^2} - \frac{6}{1 + t^2} ight) dt$ $= t - \frac{2}{t} - 6 	an^{-1} t + c$ $= \sin x - 2(\sin x)^{-1} - 6 	an^{-1} (\sin x) + c$.

$\int \frac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x} dx =$

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