Q96 / 180 · COMEDK 2026 (9 May Shift 1) · Mathematics
$\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^5 x \cos^7 x \ dx =$
- A$\dfrac{\pi}{4}$
- B$\dfrac{\pi}{2}$
- Correct answerC$0$
- D$\pi$
Solution
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 over the symmetric interval $[-\dfrac{\pi}{2}, \dfrac{\pi}{2}]$ is $0$.
$I = 0$
Answer: $0$
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 over the symmetric interval $[-\dfrac{\pi}{2}, \dfrac{\pi}{2}]$ is $0$.
$I = 0$
Answer: $0$
Hence, the correct answer is option: 'C'.