Question

Solve :

$\int \left(x^x{)}^x\right(2xlo{g}_{e}x+x)dx$

• A. $x^{\left(x^x\right)}+C$
• B. $(x^x{)}^x+C$
• C. $x^x\cdot lo{g}_{e}x+C$
• D. $\left(x^x\right)+C$

Answer 1

Answer: (B)

$(x^x{)}^x+C$

Consider the given integral.

$I=\int {\left(x^x\right)}^x(2x\log x+x)dx$

$I=\int \left(x^{x^2}\right)(2x\log x+x)dx$

Let $t=x^{x^2}$

$\log t=x^2\log x$

$\frac{1}{t}\frac{dt}{dx}=(2x\log x+x)$

$dt=x^{x^2}(2x\log x+x)dx$

Therefore,

$I=\int 1dt$

$I=t+C$

On putting the value of $t$, we get

$I=x^{x^2}+C$

$I=x^{x^x}+C$

Hence, this is the answer.