Q110 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics

$\int(\log x)^m x^n d x=$

  • A
    $\int t^m e^{n t} d t, t=e^x$
  • B
    $\int t^m e^{(n+1) t} d t, t=e^x$
  • Correct answer
    C
    $\int t^m e^{(n+1) t} d t, x=e^t$
  • D
    $\int t^m e^{n t} d t, x=e^t$

Solution

$I=\int(\log x)^m x^n d x$ $\begin{aligned} & \text { Let } \log x=t \Rightarrow x=e^t \Rightarrow d x=e^t d t \\ & I=\int t^m e^{n t} \cdot e^t d t=\int t^m e^{(n+1) t} d t\end{aligned}$

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

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