Chapter 6, Problem 17RE

To determine

To evaluate the given integral.

Answer to Problem 17RE

  ${\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\int }}\cos xdx}=\frac{\sqrt{2}}{2}$

Explanation of Solution

Given:

  ${\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\int }}\cos xdx}$

Calculation:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\int }}\cos xdx}={\left[\sin x\right]}_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\\ =\sin \left(\frac{\pi }{4}\right)-\sin \left(0\right)\\ =\frac{\sqrt{2}}{2}-0\\ =\frac{\sqrt{2}}{2}\end{array}$