First slide

Evaluation of definite integrals

Question

If $f (x) = \{\begin{array}{cc} e^{\cos x} \sin x & for | x | \leq 2 \\ 2 & otherwise , \end{array}$ then $\int_{- 2}^{3} f (x) d x =$

Easy

A

0
B

1
C

2
D

3

Solution

$\begin{array}{l} \int_{- 2}^{3} f (x) d x = \int_{- 2}^{2} f (x) d x + \int_{2}^{3} f (x) d x \\ = \int_{- 2}^{2} e^{\cos x} \sin x d x + \int_{2}^{3} 2 d x \end{array}$
$= 0 + 2 = 2 (since \int_{- a}^{a} g (x) d x = 0 if g (- x) = - g (x))$

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