Evaluate: (e^ x a +e^ a x +e^ a a ) d x - Vidyadip

MCQ

Evaluate: $\int\left(e^{x \log a}+e^{a \log x}+e^{a \log a}\right) d x$

A
$a^x \log a+\frac{x^{a+1}}{a+1}+\frac{a^a}{x}+C$
B
$a^x \log a+(a+1) x^{a+1}+a^a x+C$
✓
$\frac{a^x}{\log a}+\frac{x^{a+1}}{a+1}+a^a x+C$
D
None of these

✓

Answer

Correct option: C.

$\frac{a^x}{\log a}+\frac{x^{a+1}}{a+1}+a^a x+C$

Let $I=\int\left(e^{x \log a}+e^{a \log x}+e^{a \log a}\right) d x$
$=\int\left(e^{\log a^x}+e^{\log x^a}+e^{\log a^a}\right) d x=\int\left(a^x+x^a+a^a\right) d x$
$=\frac{a^x}{\log a}+\frac{x^{a+1}}{a+1}+a^a x+c \quad\left[\because e^{\log y=y]}\right]$

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