Question

$\int {\left(ex\right)}^x(2+\log x)dx=....+c,x\in R^{+}-\left\{1\right\}$

• A. $x^x$
• B. ${\left(ex\right)}^x$
• C. $e^x$
• D. $(1+\log x){\left(ex\right)}^x$

Answer 1

Answer: (B)

${\left(ex\right)}^x$

$\int \left(ex{)}^x\right(2+\log x)dx=\int (ex{)}^{x}dx+\int e^x.x^x(1+\log x)dx$

Integrating by parts ,

$\Rightarrow \int (ex{)}^x+e^x(\int x^x(1+\log x))-\int (\frac{d}{dx}\left(e^x\right).(\int x^x(1+\log x\left)dx\right))dx$

$\Rightarrow \int (ex{)}^x+e^x{x}^x-\int (ex{)}^x$

$\Rightarrow (ex{)}^x$