Question

If $\int \frac{\sin 2xdx}{a^2+b^{2}si{n}^{2}x}=\frac{1}{b^2}\log |f\left(x\right)|+C$, then the difference of the maximum and minimum value of $f\left(x\right)$ is

• A. $a^2$
• B. $b^2$
• C. $a^2-b^2$
• D. $a^2+b^2$

Answer 1

The correct option is B $b^2$

$I=\int \frac{\sin 2x}{a^2+b^2{\sin }^{2}x}dx;\{\begin{array}{c}\text{put}{a}^2+b^{2}si{n}^{2}x=z\\ \Rightarrow b^2\sin 2xdx=dz\end{array}$
$\Rightarrow I=\frac{1}{b^2}\int \frac{dz}{z}=\frac{1}{b^2}\ln |z|+C$
$\Rightarrow I=\frac{1}{b^2}\ln |a^2+b^2{\sin }^{2}x|+C$
So, $f\left(x\right)=a^2+b^2{\sin }^{2}x$
maximum value of $f\left(x\right)=a^2+b^2$, when $\sin x=1$
minimum value of $f\left(x\right)=a^2$, when $\sin x=0$
so difference $=(a^2+b^2)-a^2=b^2$