COMEDK · Maths · 30. Definite Integration
\(\int_\limits0^{\dfrac{\pi}{2}} \dfrac{\cos x}{1+\cos x+\sin x} d x=\)
- A \(\dfrac{\pi}{4}+\dfrac{1}{2} \log 2\)
- B \(\dfrac{\pi}{4}-\log 2\)
- C \(\dfrac{\pi}{4}+\log 2\)
- D \(\dfrac{\pi}{4}-\dfrac{1}{2} \log 2\)
Answer & Solution
Correct Answer
(D) \(\dfrac{\pi}{4}-\dfrac{1}{2} \log 2\)
Step-by-step Solution
Detailed explanation
Let \(I = \displaystyle\int_0^{\pi/2} \dfrac{\cos x}{1 + \cos x + \sin x}dx\) ... (1) Using \(\displaystyle\int_0^a f(x)dx = \int_0^a f(a-x)dx\), substitute \(x \to \dfrac{\pi}{2} - x\): \(I = \displaystyle\int_0^{\pi/2} \dfrac{\sin x}{1 + \sin x + \cos x}dx\) ... (2) Adding (1)…
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