Q114 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics

$\int(\log x)^2 d x=$

  • A
    $x \log x-2 x \log x+c$
  • B
    $x \log x+2 x \log x+c$
  • Correct answer
    C
    $x(\log x)^2-2 x(\log x-1)+c$
  • D
    $x(\log x)^2+2 x(\log x-1)+c$

Solution

Let $$ \begin{aligned} & I=\int(\log x)^2 d x \\ & I=(\log x)^2 \int d x-\int \frac{d(\log x)^2}{d x} \int d x \cdot d x+c \\ & I=x(\log x)^2-\int 2 \log x d x+c \\ & I=x(\log x)^2-2(x \log x-x)+c \\ & I=x(\log x)^2-2 x \log x+2 x+c \\ & I=x(\log x)^2-2 x(\log x-1)+c \end{aligned} $$

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

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