Question

Evaluate : $\int x^x.\left(1+\log x\right)dx$ is equal to,

• A. $x^x\log x+C$
• B. $e^{x^x}$
• C. $x^x+C$
• D. $x^{\log x}+C$

Answer 1

The correct option is C $x^x+C$

$\int x^x\left(1+\log x\right)dx$

Let $x^x=t$

Applying $log$ on both the side

$log{x}^x=logt$

$x\log x=\log t$

Taking Derivative w.r.t. $x$

$\left(x\times \frac{1}{x}+\log x\right)dx=\frac{1}{t}dt$

$\Rightarrow \left(1+\log x\right)\times x^{x}dx=dt$......$(\because t=x^x)$

$\Rightarrow \int dt$

$=t+c$

$=x^x+c$

Hence option $C$ is the answer.