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I am integrating:

$$ \int{ dx\over12+5\tan(x)} $$

I proposed $x = \arctan(u)$, replacing $dx$ by $du \over 1+u^2$, so the integral becomes:

$$ \int{du \over (1+u^2)(12+5u)} $$

Which can be integrated using partial fractions and then I eventually get that the anti-derivative is:

$$ {5 \over 169} \ln |12+5u| - {5 \over 338}\ln|1+u^2| + {12 \over 169}\arctan (u) + C $$

And finally going back to the original variable substituting $u = \tan(x)$

So my question is if this method is correct?

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    $\begingroup$ Good method and effective! $\endgroup$ – Nosrati Aug 28 '17 at 0:16
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    $\begingroup$ you did very well (+1) $\endgroup$ – tired Aug 28 '17 at 0:28
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Hint:

Multiply numerator & denominator by $\cos x$

Write $$\cos x=A(12\cos x+5\sin x)+B\cdot\dfrac{d(12\cos x+5\sin x)}{dx}$$

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