Question

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

• A. $\frac{a^{x}}{log \,a}+\frac{x^{a+1}}{a+1} + a^{a}x + c$
• B. $\frac{a^{x}}{log \,a}+\frac{x^{a+1}}{a-1} + a x^{a} + c$
• C. $\frac{a^{x}}{log\, a}+\frac{x^{a+1}}{a+1} + a x^{a} + c$
• D. $\frac{a^{x}}{log \,a}-\frac{x^{a+1}}{a+1} + a^{a} x + c$

Answer 1

Answer: (A)

We have,$ I =\int\left(e^{x\,log\,a}+e^{a\,log\,x} +e^{a\,log\,a}\right) dx $

Then,$ I =\int \left(e^{log\,a^x} +e^{log\,x^a} +e^{log\,a^a}\right)dx$

$\Rightarrow I = \int\left(a^{x} +x^{a} +a^{a}\right)dx \left[\because e^{log\lambda}=\lambda\right]$

$\Rightarrow \int a^{x}dx+\int x^{a}dx + \int a^{a}dx$

$\Rightarrow I =\frac{a^{x}}{loga}+\frac{x^{a+1}}{a+1} + a^{a } x +C$