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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Definite Integration

NDA 2016 · Mathematics · Definite Integration

What is ∫_0^ 4 π | x| dx equal to?

What is ∫04π∣cos⁡x∣dx\int_0^{4 \pi}|\cos \mathrm{x}| \mathrm{dx} equal to?

Options

  1. A0
  2. B2
  3. C4
  4. D8

Correct answer

D. 8

Step-by-step solution

∫04π∣cos⁡x∣dx=4∫0π∣cos⁡x∣dx=4[∫0π/2cos⁡xdx−∫π/2πcos⁡xdx]\begin{aligned} \int_0^{4 \pi}|\cos x| d x & =4 \int_0^\pi|\cos x| d x \\ & =4\left[\int_0^{\pi / 2} \cos x d x-\int_{\pi / 2}^\pi \cos x d x\right]\end{aligned}=4[(sin⁡x)0π/2−(sin⁡x)π/2π]=4[sin⁡π2−0−sin⁡π+sin⁡π2]=4[2sin⁡π2]=8\begin{aligned} & =4\left[(\sin x)_0^{\pi / 2}-(\sin x)_{\pi / 2}^\pi\right] \\ & =4\left[\sin \frac{\pi}{2}-0-\sin \pi+\sin \frac{\pi}{2}\right] \\ & =4\left[2 \sin \frac{\pi}{2}\right]=8\end{aligned}$
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