Q87 / 180 · COMEDK 2012 · Mathematics

$\int x^{x}(1+\log x) d x=$

  • Correct answer
    A
    $x^{x}+C$
  • B
    $x^{-x}+x$
  • C
    $x \log x+x$
  • D
    $\log x+x$

Solution

We have, $I=\int x^{x}(1+\log x) d x$ Let $\quad x^{x}=t$ $\Rightarrow \quad \frac{d t}{d x}=x^{x} \log x+x^{x}$ $\Rightarrow \quad \frac{d t}{d x}=x^{x}(1+\log x)$ $\Rightarrow \quad d t=x^{x}(1+\log x) d x$ Now, $\quad I=\int d t$ $$ =t+C=x^{x}+C $$

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

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