Q. $\int_{}^{}{\left\lbrack e^{a\log x} + e^{x\log a} \right\rbrack dx = ... + c;a,x > 1}$
A) $\frac{x^{a + 1}}{a + 1} + \frac{a^{x}}{\log a}$
B) $\frac{e^{a\log x}}{a\log x} + \frac{e^{x\log a}}{x\log a}$
C) $\frac{x^{a - 1}}{a - 1} + a^{x}.\log a$
D) $\frac{e^{a\log x}}{\frac{a}{x}} + \frac{e^{x\log a}}{x}$
Explanation
(a)
Let, $I = \int_{}^{}{\left\lbrack e^{a\log x} + e^{x\log a} \right\rbrack dx}$
$= \int_{}^{}{\left\lbrack e^{\log_{e}x^{a}} + e^{\log_{e}a^{x}} \right\rbrack dx}$
$= \int_{}^{}{\left\lbrack x^{a} + a^{x} \right\rbrack dx} = \frac{x^{a + 1}}{a + 1} + \frac{a^{x}}{\log a} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.