Question

Use the substitution formula to evaluate the definite integral. See attachment.

Answer 1

As it suggests, use the substitution u=cos(x). This means that
\[du = -\sin(x) dx\]Also, substituting for the integral limits we find that
\[x=\pi/3 =>u = 1/2\]and
\[x=2\pi =>u=1\]Therefore the integral becomes
\[-3\int\limits_{1/2}^{1}u ^{2}du\]which can be calculated in the usual way.