Question

Evaluate : $\int \left(e^{xloga}+e^{alogx}+e^{aloga}\right)dx$

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

Answer 1

Answer: (A)

We have,$I=\int \left(e^{xloga}+e^{alogx}+e^{aloga}\right)dx$

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

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

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

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