Question:
Evaluate:
\[
\int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4\,dx
\]
Evaluate:
\[
\int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4\,dx
\]
Show Hint
For large trigonometric powers, convert everything into powers of \(\sin\frac{x}{2}\) and \(\cos\frac{x}{2}\).
Updated On: Jun 17, 2026
- \(0\)
- \(5\pi\)
- \(\dfrac{5\pi}{2}\)
- \(\dfrac{5\pi}{4}\)
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
Use half-angle identities to simplify powers of trigonometric expressions.
Step 1: Apply identities. Recall, \[ 1+\cos x=2\cos^2\frac{x}{2} \] and \[ 1-\cos x=2\sin^2\frac{x}{2} \] Hence, \[ (1+\cos x)^3(1-\cos x)^4 = (2\cos^2\tfrac{x}{2})^3 (2\sin^2\tfrac{x}{2})^4 \] \[ = 2^7\cos^6\frac{x}{2}\sin^8\frac{x}{2} \] Using symmetry and standard integral evaluation, this reduces to \[ \int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4dx = \frac{5\pi}{4} \] Therefore, \[ \boxed{\frac{5\pi}{4}} \]
Step 1: Apply identities. Recall, \[ 1+\cos x=2\cos^2\frac{x}{2} \] and \[ 1-\cos x=2\sin^2\frac{x}{2} \] Hence, \[ (1+\cos x)^3(1-\cos x)^4 = (2\cos^2\tfrac{x}{2})^3 (2\sin^2\tfrac{x}{2})^4 \] \[ = 2^7\cos^6\frac{x}{2}\sin^8\frac{x}{2} \] Using symmetry and standard integral evaluation, this reduces to \[ \int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4dx = \frac{5\pi}{4} \] Therefore, \[ \boxed{\frac{5\pi}{4}} \]
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