Question

ᶴ(cos x/3sinx+4√(sin x))
please i need procedure how to integrate this function

Answer 1

lets see , try u sub then partial fractions

u = sin x
du = cos x dx

that will get rid of the cos x on top
then
\[\int\limits \frac{du}{3u + 4\sqrt{u}} = \int\limits \frac{du}{\sqrt{u} (3 \sqrt{u} +4)}\]
\[\rightarrow \frac{A}{\sqrt{u}} + \frac{B}{3\sqrt{u} +4} = \frac{1}{\sqrt{u}(3 \sqrt{u} +4)}\]
3A +B = 0
4A = 1 --> A = 1/4
--> B = -3/4
\[\frac{1}{4} \int\limits \frac{du}{\sqrt{u}} - \frac{3}{4} \int\limits \frac{du}{3 \sqrt{u} + 4}\]