$$ \int \frac{d x}{12 \cos x+5 \sin x}= $$
$\frac{1}{13} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}-\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$
$\frac{5}{12} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}-\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$
$\frac{1}{13} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}+\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$
$\frac{5}{12} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}+\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$
If $\int \frac{\cos ^3 x}{\sin ^2 x+\sin ^4 x} d x=c-\operatorname{cosec} x-f(x)$, then $f\left(\frac{\pi}{2}\right)=$
1
0
$\pi / 2$
$\pi$
$$ \int \frac{13 \cos 2 x-9 \sin 2 x}{3 \cos 2 x-4 \sin 2 x} d x= $$
$3 x-\frac{1}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$
$\frac{x}{2}-3 \log |3 \cos 2 x-4 \sin 2 x|+C$
$3 x+\frac{1}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$
$x+\frac{3}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$
$$ \int \sqrt{x^2+x+1} d x $$
$\frac{(2 x+1)}{4} \sqrt{x^2+x+1}+\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$
$\frac{x+1}{4} \sqrt{x^2+x+1}+\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$
$\frac{x+1}{4} \sqrt{x^2+x+1}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$
$\frac{(2 x+1)}{4} \sqrt{x^2+x+1}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$
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