If $$f(x)=$$$$cos$$⁡x and $$g(x)=$$$$sin$$⁡x then ∫log⁡$$f(x)(f(x))2dx$$ is

Correct option is 4g(x)f(x)[log⁡f(x)+1]−x+C

Questions

If $f (x) = \cos x$ and $g (x) = \sin x$ then $\int \frac{\log f (x)}{(f (x))^{2}} d x$ is

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f(x)(log⁡f(x)+1)+x+C

g(x)f(x)(log⁡f(x)+1)+x22+C

f(x)g(x)(log⁡g(x)+1)+x+C

g(x)f(x)[log⁡f(x)+1]−x+C

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detailed solution

Correct option is D

I=∫log⁡cos⁡xcos2⁡xdx=∫sec2⁡xlog⁡cos⁡xdx=(tan⁡x)log⁡cos⁡x−∫tan⁡x−sin⁡xcos⁡xdx=(tan⁡x)log⁡cos⁡x+∫sec2−1dx=(tan⁡x)[log⁡cos⁡x+1]−x+C=g(x)f(x)[log⁡f(x)+1]−x+C

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If f(x)=cos⁡x and g(x)=sin⁡x then ∫log⁡f(x)(f(x))2dx is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $f (x) = \cos x$ and $g (x) = \sin x$ then $\int \frac{\log f (x)}{(f (x))^{2}} d x$ is