Question

Solve $\int \frac{\sin x\cos x}{5{\sin }^{2}x+3{\cos }^{2}x}dx$.

Answer 1

We are given, $I=\int \frac{\sin x.\cos x}{5{\sin }^{2}x+3{\cos }^{2}x}dx$

$I=\int \frac{\sin x.\cos x}{5{\sin }^{2}x+3(1-{\sin }^{2}c)}dx$

(as we know ${\sin }^2\theta +{\cos }^2\theta =1$)

$=\int \frac{\sin x.\cos x.dx}{5{\sin }^{2}x+3{\sin }^{2}x}$

$=\int \frac{\sin x.\cos xdx}{3+2{\sin }^{2}x}$

Let ${\sin }^{2}x=t$, differentiating it with respect to $x$ we get

$23\sin x\cos xdx=dt$

$\therefore \sin x\cos xdx=\frac{dt}{2}$

$\int \frac{dt}{2(3+2t)}$

$=\int \frac{dt}{6+4t}$

$I=\frac{1}{4}\log |6+4t|+C$

$I=\int \frac{\sin x.\cos x}{5{\sin }^{2}x+3{\cos }^{2}x}=\frac{1}{4}\log |6+4{\sin }^{2}x|+C$