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Question Number 172608 by Mikenice last updated on 29/Jun/22

Answered by FelipeLz last updated on 03/Jul/22

$$f\left(x\right)=x-\sin \left(x\right)$$$$\int \frac{x\cos \left(x\right)-\sin \left(x\right)}{{[x-\sin \left(x\right)]}^2}dx$$$$\int \frac{x\cos \left(x\right)-x+x-\sin \left(x\right)}{{[x-\sin \left(x\right)]}^2}dx$$$$\int \frac{[x-\sin \left(x\right)]-x[1-\cos \left(x\right)]}{{[x-\sin \left(x\right)]}^2}dx$$$$\int \frac{f\left(x\right)-{xf}^{\prime }\left(x\right)}{{\left[f\left(x\right)\right]}^2}dx=\frac{x}{f\left(x\right)}+c=\frac{x}{x-\sin \left(x\right)}+c$$

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