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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2010 · Mathematics · Indefinite Integration

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

What is equal to? ∫ex(1+x)cos⁡2(xex)dx\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x where cc is a constant of integration.

Options

  1. Axex+cx e^x+c
  2. Bcos⁡(xex)+c\cos \left(x e^x\right)+c
  3. Ctan⁡(xex)+c\tan \left(x e^x\right)+c
  4. Dxcosec⁡(xex)+cx \operatorname{cosec}\left(x e^x\right)+c

Correct answer

C. tan⁡(xex)+c\tan \left(x e^x\right)+c

Step-by-step solution

∫ex(1+x)cos⁡2(xex)dxPut xex=z∴ex(1+x)dx=dz∫dzcos⁡2z=∫sec⁡2zdz=tan⁡z+c=tan⁡(xex)+c\begin{aligned} & \int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x \\ & \text{Put } x e^x=z \\ & \therefore e^x(1+x) d x=d z \\ & \int \frac{d z}{\cos ^2 z}=\int \sec ^2 z d z=\tan z+c \\ & =\tan \left(x e^x\right)+c\end{aligned}
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