# Is this method of indefinite integration correct? $\int{ dx\over12+5\tan(x)}$

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?

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

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