Q250 / 270 · NDA 2021 (Phase 1) · Mathematics

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

  • A
    $\frac{x^4}{4}+c$
  • B
    $\frac{x^3}{3}+c$
  • C
    $\frac{2 x^5}{5}+c$
  • Correct answer
    D
    $\frac{x^5}{5}+c$

Solution

$\begin{aligned} & \mathrm{I}=\int e^{\left(2 \ln x+\ln x^2\right)} \cdot d x \\ & =\int e^{\left(\ln x+\ln x^2\right)} \cdot d x=\int e^{\left(2 \cdot \ln x^2\right)} \cdot d x\end{aligned}$ $=\int e^{\left(\ln x^4\right)} \cdot d x=\int x^4 \cdot d x=\frac{x^5}{5}+c$

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

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