Question

\({\bf{1 - 40}}\)- Evaluate the integral.

\(\int_0^{\pi /2} {\sin } \theta {e^{\cos \theta }}d\theta \)

Short answer

\(\int\limits_0^{\frac{\pi }{2}} {\sin \theta {e^{\cos \theta }}} d\theta = e - 1\)

Step-by-step solution

Definition

Integrationis a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation.

Substitute

Let\(y = \cos \theta \)

\( \Rightarrow dy = - \sin \theta d\theta \)

Substituting these values into the expression inside the integral gives

\(\begin{aligned}{c}\sin \theta {e^{\cos \theta }} = {e^u}( - du)\\ = - {e^u}du\end{aligned}\)

And limit changes to:

\(\begin{aligned}{c}y = \cos (0)\\ &= 1\end{aligned}\) and\(\begin{aligned}{c}y &= \cos (\frac{\pi }{2})\\ &= 0\end{aligned}\)

Evaluate integral

Substituting values & applying new limits we have:

\(\begin{aligned}{c}\int_1^0 - {e^n}dy &= \left( { - {e^n}} \right)_1^0\\ &= - 1 + e\\ &= e - 1\end{aligned}\)