Q. If $f(x) = \lim_{y \rightarrow x}\frac{\sin^{2}{y - \sin^{2}x}}{y^{2} - x^{2}}$, then $\int_{}^{}{4x\ f(x)dx}$ is equal to
A) $\cos{2x + c}$
B) $2\cos{2x + c}$
C) $- \cos{2x + c}$
D) $- 2\cos{2x + c}$
Explanation
(c)
$f(x) = \lim_{y \rightarrow x}\frac{\sin^{2}{y - \sin^{2}x}}{y^{2} - x^{2}}$ $\left\lbrack \frac{0}{0}\ form \right\rbrack$
$= \lim_{y \rightarrow x}\frac{2\sin{y\cos{y - 0}}}{2y - 0}$
$= \frac{\sin{2x}}{2x}$
$\therefore\int_{}^{}{4x}\ f(x)dx = \int_{}^{}{4x}\ \left( \frac{\sin{2x}}{2x} \right)dx$
$= 2\int_{}^{}{\sin{2x}}\ dx$
$= - \cos{2x + c}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.