COMEDK · Maths · 30. Definite Integration
\(\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^5 x \cos^7 x \ dx =\)
- A \(\dfrac{\pi}{4}\)
- B \(\dfrac{\pi}{2}\)
- C \(0\)
- D \(\pi\)
Answer & Solution
Correct Answer
(C) \(0\)
Step-by-step Solution
Detailed explanation
Let \(I = \displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \sin^5 x \cos^7 x \ dx\) Let \(f(x) = \sin^5 x \cos^7 x\) \(f(-x) = \sin^5(-x) \cos^7(-x)\) \(f(-x) = (-\sin x)^5 (\cos x)^7\) \(f(-x) = -\sin^5 x \cos^7 x = -f(x)\) Since \(f(x)\) is an odd function, the integral…
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