How can we find the indefinite integral for:
$$\int \frac{\sqrt{\cot(x)} – \sqrt{\tan(x)}}{4+3 \sin^2 (x)} \ \mathrm d x$$
I tried expressing the denominator in $\sin x+\cos x$ form since we have its derivative in numerator but I am not able to proceed. Please help me out.
Best Answer
By some basic trigonometric identities, one has $$\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)}~dx=\int \frac{\sec^2(x)(1-\tan(x))}{\sqrt{\tan(x)}(7\tan^2(x)+4)}~dx,$$ which motivates the substitution $u=\tan(x)$. This leads to $$\int \frac{1-u}{\sqrt{u}(7u^2+4)}~du,$$ whose integrand can be converted to a rational expression via the substitution $u=v^2$. The resulting expression can be solved using partial fractions.