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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2022 · Mathematics · Indefinite Integration

What is ∫ e^ x (1+ x+x x) d x equal to?

What is ∫ex(1+ln⁡x+xln⁡x)dx\int e^{\mathrm{x}}(1+\ln x+x \ln x) d x equal to?

Options

  1. Axexln⁡x+cx e^{\mathrm{x}} \ln x+\mathrm{c}
  2. Bx2exln⁡x+cx^2 e^{\mathrm{x}} \ln x+\mathrm{c}
  3. Cx+exln⁡x+cx+e^{\mathrm{x}} \ln x+\mathrm{c}
  4. Dxex+ln⁡x+cx e^{\mathrm{x}}+\ln x+\mathrm{c}

Correct answer

A. xexln⁡x+cx e^{\mathrm{x}} \ln x+\mathrm{c}

Step-by-step solution

∵d(xln⁡x)dx=1+ln⁡x∴I=∫ex{1+ln⁡x+xln⁡x}dx=xln⁡xex+c[∵∫ex(f(x)+f′(x)dx=exf(x)+c)]\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}
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