Question

Evaluate:$\int \frac{1}{4\cos x+3\sin x}dx$

Answer 1

$\int \frac{dx}{4\cos x+3\sin x}$

$\int \frac{dx}{4\left(\frac{1-{\tan }^{2}x/2}{1+{\tan }^{2}x/2}\right)+3\left(\frac{2\tan x/2}{1+{\tan }^{2}x/2}\right)}$

$\int \frac{(1+{\tan }^{2}x/2)dx}{4-4{\tan }^{2}x/2+6\tan x/2}$

Let $\tan x/2=t$

${\sec }^2\frac{x}{2}dx=2dt$

$\int \frac{2dt}{4-4t^2+6t}$

$\int \frac{dt}{2-2t^2+3t}$

$\frac{1}{2}\int \frac{dt}{1-t^2+\frac{3}{2}t}$

$\frac{1}{2}\int \frac{dt}{1-(t^2-\frac{3}{2}t)}$

$\frac{1}{2}\int \frac{dt}{1-{(t^2-\frac{3t}{2}+t{\left(\frac{3}{4}\right)}^2)}^2+{\left(\frac{3}{4}\right)}^2}$

$\frac{1}{2}\int \frac{dt}{1+\frac{9}{16}-(t-3/2{)}^2}$

$\frac{1}{2}\int \frac{dt}{25/16-(t-3/2{)}^2}$

$\frac{1}{2}\int \frac{dt}{(5/4{)}^2-(t-3/2{)}^2}$

$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C$

$\frac{1}{2}\frac{1}{2\left(5/4\right)}\ln \left|\frac{5/4+t-3/2}{5/4-t+3/2}\right|+C$

$\frac{1}{5}\ln \left|\frac{t-1/4}{11/4-t}\right|+C$

$\frac{1}{5}\ln \left|\frac{4t-1}{11-4t}\right|+C$

$\frac{1}{5}\ln \left|\frac{4\tan x/2-1}{11-4\tan x/2}\right|+C$