Q224 / 270 · NDA 2022 (Phase 2) · Mathematics

What is $\int\left(x^x\right)^2(1+\ln x) d x$ equal to?

  • A
    $x^{2 \mathrm{x}}+c$
  • Correct answer
    B
    $\frac{1}{2} x^{2 \mathrm{x}}+c$
  • C
    $2 x^{2 \mathrm{x}}+c$
  • D
    $\frac{1}{2} x^x+c$

Solution

$I=\int\left(x^x\right)^2\left(1+\int \ln x\right) d x$ Let $x^{\mathrm{x}}=t$ $\begin{aligned} & \Rightarrow \ln t=x \ln x \Rightarrow d t=x^{\mathrm{x}}(1+\ln x) d x \\ & I=\int t d t=\frac{-}{2}+C=-(x)+C \\ & \frac{1}{2} x^{2 \mathrm{x}}+C \end{aligned}$

Hence, the correct answer is option: 'B'.

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