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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2019 · Mathematics · Indefinite Integration

What is ∫ (x^2) d x equal to ?

What is ∫ln⁡(x2)dx\int \ln \left(x^2\right) d x equal to ?

Options

  1. A2xln⁡(x)−2x+c2 \mathrm{x} \ln (\mathrm{x})-2 \mathrm{x}+\mathrm{c}
  2. B2x+c\frac{2}{x}+c
  3. C2xln⁡(x)+c2 \mathrm{x} \ln (\mathrm{x})+\mathrm{c}
  4. D2ln⁡(x)x−2x+c\frac{2 \ln (x)}{x}-2 x+c

Correct answer

A. 2xln⁡(x)−2x+c2 \mathrm{x} \ln (\mathrm{x})-2 \mathrm{x}+\mathrm{c}

Step-by-step solution

∫ln⁡(x)⋅dx2⋅∫ln⁡x⋅dx\int \ln (x) \cdot d x \quad 2 \cdot \int \ln x \cdot d x=2∫1⋅ln⁡x⋅dx=2[ln⁡x⋅x−∫1x⋅x⋅dx=2(x⋅ln⁡x−x)+c=2xln⁡x−2x+c\begin{aligned} & =2 \int 1 \cdot \ln x \cdot d x \\ & =2\left[\ln x \cdot x-\int \frac{1}{x} \cdot x \cdot d x\right. \\ & =2(x \cdot \ln x-x)+c \\ & =2 x \ln x-2 x+c\end{aligned}$$
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