Question

Evaluate $\int \frac{a^{\sqrt{x}}}{\sqrt{x}}dx$

• A. $\frac{2a^{\sqrt{x}}}{\log a}+c$
• B. $\frac{\log a}{a^{\sqrt{x}}}+c$
• C. $2a^{\sqrt{x}}\log a+c$
• D. $a^{\sqrt{x}}{\log }_{a}e+c$

Answer 1

The correct option is A $\frac{2a^{\sqrt{x}}}{\log a}+c$

$\int \frac{a^{\sqrt{x}}}{\sqrt{x}}dx$

put $\sqrt{x}=t$

$\frac{1}{2\sqrt{x}}dx=dt$

$\frac{dx}{\sqrt{x}}=2dt$

$\int a^t.2dt=2\int a^{t}dt$

$=2\frac{a^t}{\log a}+c$

$=\frac{2a^{\sqrt{x}}}{\log a}+c$