Question

Compute the definite integral of the following function over the interval \( [0, \pi ] \)
\[
f(x)=\sin ^{2}(x)+\cos ^{2}(x)
\]

Answer 1

The function \(f(x)=\sin ^2(x)+\cos ^2(x)\) is a well-known trigonometric identity that is always equal to 1 for all \(x\). Therefore, the integral of this function over any interval is simply the length of the interval.
So, the definite integral of \(f(x)\) from 0 to \(\pi\) is:
\[
\int_0^\pi f(x) d x=\int_0^\pi 1 d x=[\pi-0]=\pi
\]
So, the definite integral of \(f(x)=\sin ^2(x)+\cos ^2(x)\) from 0 to \(\pi\) is \(\pi\).