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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2015 · Mathematics · Indefinite Integration

What is ∫ (x e^x d x)/((x+1)^2) equal to? where c is the constant...

What is ∫xexdx(x+1)2\int \frac{x e^x d x}{(x+1)^2} equal to? where cc is the constant integration.

Options

  1. A(x+1)2ex+c(x+1)^2 e^x+c
  2. B(x+1)ex+c(x+1) e^x+c
  3. Cexx+1+c\frac{e^x}{x+1}+c
  4. Dex(x+1)2+c\frac{e^x}{(x+1)^2}+c

Correct answer

C. exx+1+c\frac{e^x}{x+1}+c

Step-by-step solution

∫xex(1+x)2dx=∫ex[11+x−1(1+x)2]dx=ex1+x+C{∫ex(f(x)+f′(x))dx=exf(x)}\begin{aligned} & \int \frac{x e^x}{(1+x)^2} d x \\ & =\int e^x\left[\frac{1}{1+x}-\frac{1}{(1+x)^2}\right] d x \\ & =\frac{e^x}{1+x}+C \quad\left\{\int e^x\left(f(x)+f^{\prime}(x)\right) d x=e^x f(x)\right\}\end{aligned}
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