Q. The value of $\int_{}^{}{x\left( x^{x} \right)^{x}}\left( 2\log{x + 1} \right)dx$ is
A) $\left( x^{x} \right)^{x} + c$
B) $x^{x} + c$
C) $x^{\log x} + c$
D) None of these
Explanation
(a)
Let $I = \int_{}^{}x\left( x^{x} \right)^{x}\left( 2\log{x + 1} \right)dx$
Let $\left( x^{x} \right)^{x} = t,\ x^{2}\log x = \log t$
$\Rightarrow \left( x^{2}.\frac{1}{x} + 2x\log x \right)dx = \frac{1}{t}dt$
$\Rightarrow x\left( 1 + 2\log x \right)dx = \frac{1}{t}\ dt$
$\therefore\ I = \int_{}^{}t.\frac{1}{t}\ dt = t + c = \left( x^{x} \right)^{x} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.