Q. If $\int_{}^{}{f(x)dx = F(x)},\ then\ \int_{}^{}{x^{3}f\left( x^{2} \right)dx}$ is equal to
A) $\frac{1}{2}\{ x^{2}F\left( x^{2} \right) - \int_{}^{}{F\left( x^{2} \right)dx^{2}\}\ }$
B) $\frac{1}{2}\{ x^{2}F\left( x^{2} \right) - \int_{}^{}{F\left( x^{2} \right)dx\}\ }$
C) $\frac{1}{2}\left( x^{2}F(x) - \frac{1}{2}\int_{}^{}{F\left( x^{2} \right)dx} \right)$
D) None of the above
Explanation
(a)
We have, $\int_{}^{}{f(x)dx = F(x)}$ ...(i)
Let $I = \int_{}^{}{x^{3}f\left( x^{2} \right)dx}$
Put $x^{2} = t\ \Rightarrow 2x\ \ dx = dt$
$\therefore I = \frac{1}{2}\int_{}^{}{t\ f(t)dt}$
$= \frac{1}{2}\{\ t\int_{}^{}{f(t)dt - \int_{}^{}{(\ \int_{}^{}{f\ (t)dt)\ dt\ \}\ }}}$
$= \frac{1}{2}\{\ tF(t) - \int_{}^{}{F(t)dt\ \}}$ [From Eq. (i)]
$= \frac{1}{2}\{\ x^{2}F\left( x^{2} \right) - \int_{}^{}{F\left( x^{2} \right)dx^{2}\}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.