The value of \(\int_{0}^{\frac{\pi}{2}}\displaystyle\left(\sin^{4}x-\cos^{4}x\right)dx\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0
Factorising as a difference of squares,
\(\sin^{4}x-\cos^{4}x = (\sin^{2}x-\cos^{2}x)(\sin^{2}x+\cos^{2}x)\)
\( = \sin^{2}x-\cos^{2}x = -\cos 2x\)
So
\( \displaystyle \int_{0}^{\pi/2}\left(\sin^{4}x-\cos^{4}x\right)dx = -\left[\frac{\sin 2x}{2}\right]_{0}^{\pi/2} \)
\( \displaystyle = -\left(\frac{\sin\pi}{2} - \frac{\sin 0}{2}\right) = 0 \)