Mathematics

Evaluate $$\int {{e^x}\left( {\tan x - \log \,\cos \,x} \right)\,dx} $$


SOLUTION
It is just an observation to make perfect differential;
it is in the form of  $$\displaystyle \int (e^{x}(f(x))+e^{x}(\dfrac{d}{dx}f(x)))dx=e^{x}(f(x))$$
$$\dfrac{d}{dx}\log(\cos x)=-\tan x$$
so here $$f(x)=\log(\cos x)$$
so,
              $$\displaystyle\int e^{x}(\tan x-\log \cos x)dx=e^{x}(-\log\cos x)$$
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Subjective Medium Published on 17th 09, 2020
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  • A. $$\displaystyle \frac{4}{3}\log \left ( 2x^{2}+x+1 \right )+\frac{1}{2\sqrt{\left ( 7 \right )}}\tan ^{-1}\frac{3x+1}{\sqrt{\left ( 7 \right )}}.+C$$
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