Question

$\int \frac{e^{x-1}+x^{e-1}}{\sqrt{e^x+x^e}}dx=$

• A. $\frac{2}{e}\sqrt{e^x+x^e}+c$
• B. $2\sqrt{e^x+x^e}+c$
• C. $\frac{\sqrt{e^x+x^e}}{e}+c$
• D. $\frac{2}{3e}\sqrt{e^x+x^e}+c$

Answer 1

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

$\int \frac{e^{x-1}+x^{e-1}}{\sqrt{e^x+x^e}}dx$

Substitute $e^x+x^e=t$

$\Rightarrow (e^x+e{x}^{e-1})dx=dt$

$=e(e^{x-1}+x^{e-1})dx=dt$

$=\frac{1}{e}\int \frac{dt}{\sqrt{t}}=\frac{2}{e}\sqrt{t}+c$

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