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Question Number 211885 by Spillover last updated on 23/Sep/24

Answered by Frix last updated on 23/Sep/24

$$\int \frac{x^2}{{(x\sin x+\cos x)}^2}dx=\frac{y}{x\sin x+\cos x}+C$$$$\frac{d}{dx}\left[\frac{y}{x\sin x+\cos x}\right]=\frac{x^2}{{(x\sin x+\cos x)}^2}$$$$\frac{y^{\prime }(x\sin x+\cos x)-yx\cos x}{{(x\sin x+\cos x)}^2}=\frac{x^2}{{(x\sin x+\cos x)}^2}$$$$\Rightarrow y=\sin x-x\cos x$$$$\Rightarrow$$$$\int \frac{x^2}{{(x\sin x+\cos x)}^2}dx=\frac{\sin x-x\cos x}{x\sin x+\cos x}+C$$

Commented by Spillover last updated on 24/Sep/24

$$thanks$$

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