JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of \(\int_0^{20\pi} (\sin^4 x + \cos^4 x) \, dx\) is equal to:
- A \(\dfrac{15\pi}{2}\)
- B \(25\pi\)
- C \(15\pi\)
- D \(\dfrac{25\pi}{2}\)
Answer & Solution
Correct Answer
(C) \(15\pi\)
Step-by-step Solution
Detailed explanation
Let \(I = \int_0^{20\pi} (\sin^4 x + \cos^4 x) \, dx\) Using the algebraic identity \(a^2 + b^2 = (a+b)^2 - 2ab\), the integrand can be simplified: \(\sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x\) Since \(\sin^2 x + \cos^2 x = 1\) and…
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