\(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\displaystyle \sin^7 x\,dx =\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 0
Let \(f(x) = \sin^7 x\). Then
\(f(-x) = \left(\sin(-x)\right)^7\)
\(= (-\sin x)^7\)
\(= -\sin^7 x = -f(x)\)
so f is an odd function. For an odd function, \(\int_{-a}^{a} f(x)\,dx = 0\), because the areas on the two sides of the origin cancel. Hence
\( \displaystyle \int_{-\pi/2}^{\pi/2}\sin^7 x\,dx = 0 \)
NCERT (2026-27 edition):
Some Properties of Definite Integrals
