\( \int_{-\pi/3}^{\pi/3}\displaystyle 2x^5\cos^4x\,dx \) equals
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
The integral of an odd function over limits symmetric about 0 is zero: \( \int_{-a}^{a}f(x)\,dx=0 \) when \( f(-x)=-f(x) \).
\( \displaystyle f(x)=2x^5\cos^4x \)
\( \displaystyle f(-x)=2(-x)^5\cos^4(-x) \)
\( \displaystyle =-2x^5\cos^4x=-f(x) \)
So f is odd and the limits are symmetric about 0:
\( \displaystyle \int_{-\pi/3}^{\pi/3}2x^5\cos^4x\,dx=0 \)