Question

$\int \frac{sin2x}{a{\cos }^{2}x+b{\sin }^{2}x}dx=$

• A. $\frac{1}{b+a}\log |aco{s}^{2}x+bsi{n}^{2}x|+C$
• B. $\frac{1}{b-a}\log |aco{s}^{2}x-bsi{n}^{2}x|+C$
• C. $\frac{1}{b-a}\log |aco{s}^{2}x+bsi{n}^{2}x|+C$
• D. $\frac{1}{b+a}\log |aco{s}^{2}x-bsi{n}^{2}x|+C$

Answer 1

The correct option is C $\frac{1}{b-a}\log |aco{s}^{2}x+bsi{n}^{2}x|+C$

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

$\Rightarrow \int \frac{\sin 2x}{(b-a){\sin }^{2}x+a}dx$

putting ${\sin }^{2}x.m$

$2sinxcosxdx=dm$

${\sin }^{2}xdx=dm$

$\Rightarrow \int \frac{dm}{(b-a)m+a}$

$\Rightarrow \frac{1}{b-a}In|(b-a)m+a|+c$

$\Rightarrow \frac{1}{b-a}In|(b-a){\sin }^{2}x+a|+c$

$\Rightarrow \frac{1}{b-a}In|b{\sin }^{2}x+a{\cos }^{2}x|+c$