COMEDK · Maths · 28. Indefinite Integration
\(\int \left(\sin^6 x + \cos^6 x + 3\sin^2 x \cos^2 x\right)\, dx =\)
- A \(-\dfrac{3}{2}\cos 2x + C\)
- B \(\dfrac{2}{3}x + C\)
- C \(\dfrac{3}{2}\sin 2x + C\)
- D \(x + C\)
Answer & Solution
Correct Answer
(D) \(x + C\)
Step-by-step Solution
Detailed explanation
Let \(I = \int (\sin^6 x + \cos^6 x + 3\sin^2 x \cos^2 x)\, dx\) Using the algebraic identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\), we can write: \(\sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)^3 - 3\sin^2 x \cos^2 x (\sin^2 x + \cos^2 x)\) Since \(\sin^2 x + \cos^2 x = 1\), we get:…
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