Question

Solve $\frac{sin2x}{aco{s}^{2}x+bsi{n}^{2}x}$

Answer 1

Consider the given function:

$\int \frac{\sin 2x}{a{\cos }^{2}x+b{\sin }^{2}x}dx$

$puta{\cos }^{2}x+b{\sin }^{2}x=t$

$(-a.2\cos x\sin x+b.2\sin x\cos x)\frac{dx}{dt}=1$

$(b-a).\sin 2xdx=dt$

$\sin 2xdx=\frac{1}{b-a}dt$

$\int \frac{\sin 2x}{a{\cos }^{2}x+b{\sin }^{2}x}dx$

$=\frac{1}{b-a}\int \frac{1}{t}dt$

$=\frac{1}{b-a}\log t+c$

$=\frac{1}{b-a}\log (a{\cos }^{2}x+b{\sin }^{2}x)+c$

Hence this is the answer.

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