Question

Why can \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\] be written as \[2\int_{0}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?

Options

• \[\sin^{2}x\] satisfies \[f(2a-x)=-f(x)\]

• \[\sin^{2}x\] is an even function

• The upper and lower limits are equal

• \[\sin^{2}x\] is an odd function

Answer 1

\[\sin^{2}x\] is an even function. Therefore \[P_7\](i) changes the symmetric integral into twice the integral from \[0\] to \[\frac{\pi}{4}\].