# Trigonometric Integral : $\int\frac{1}{\sin x+ 3\cos x}dx$ [closed]

I would appreciate if somebody could help me with the following problem

Q: How to integrate this integral $$\int\frac{1}{\sin x+ 3\cos x}dx$$

• Perhaps you could use a more informative title? – davidlowryduda Dec 10 '13 at 1:44
• Perhaps by using the tangent half-angle substitution ? – Lucian Dec 10 '13 at 1:54
• @Lucian that's the answer to every question :) – Igor Rivin Dec 10 '13 at 2:09
• @IgorRivin: Even to this one ? :-) – Lucian Dec 10 '13 at 2:42
• @Lucian especially this one... – Igor Rivin Dec 10 '13 at 2:58

Switching $y=\tan \frac x 2 \implies\frac {dy}{dx} = \frac 1 2 \frac 1 {\cos^2 \frac x 2}$
$$1\cdot \sin x +3 \cdot \cos x=\sqrt{10} \cdot \sin (x+a)$$ where $a=\tan ^{-1}(3)$
So, now can you just apply the known integral of $\csc(x+a)$?