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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Definite Integration

NDA 2024 · Mathematics · Definite Integration

What is ∫_0^ (π)/(2) (a+ x)/(2 a+ x+ x) d x equal to?

What is ∫0π2a+sin⁡x2a+sin⁡x+cos⁡xdx\int_0^{\frac{\pi}{2}} \frac{a+\sin x}{2 a+\sin x+\cos x} d x equal to?

Options

  1. Aπ4\frac{\pi}{4}
  2. Bπ2\frac{\pi}{2}
  3. C1
  4. D0

Correct answer

A. π4\frac{\pi}{4}

Step-by-step solution

Let, \quad\mathrm{I}=∫0π2a+sin⁡x2a+sin⁡x+cos⁡xdx=\int_0^{\frac{\pi}{2}} \frac{a+\sin x}{2 a+\sin x+\cos x} d x\qquad ....(i) I=∫0π2a+cos⁡x2a+cos⁡x+sin⁡xdx\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{a+\cos x}{2 a+\cos x+\sin x} d x\qquad ....(ii) Adding equation (i) and (ii) 2I=∫0π22a+sin⁡x+cos⁡x2a+sin⁡x+cos⁡xdx=∫0π2dx=π2⇒I=π4\begin{aligned} 2 \mathrm{I} & =\int_0^{\frac{\pi}{2}} \frac{2 a+\sin x+\cos x}{2 a+\sin x+\cos x} d x \\ & =\int_0^{\frac{\pi}{2}} d x=\frac{\pi}{2} \\ \Rightarrow \quad \mathrm{I} & =\frac{\pi}{4} \end{aligned}
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