Q225 / 270 · NDA 2022 (Phase 2) · Mathematics

What is $\int e^{\mathrm{x}}(1+\ln x+x \ln x) d x$ equal to?

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

Solution

$\begin{aligned} & \because \frac{d(x \ln x)}{d x}=1+\ln x \\ & \therefore I=\int e^x\{1+\ln x+x \ln x\} d x \\ & =x \ln x e^{\mathrm{x}}+c\left[\because \int e^x\left(f(x)+f^{\prime}(x) d x=e^x f(x)+c\right)\right]\end{aligned}$

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

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