Indefinite Integration 1 Question 27

28. If $f(x)=\begin{array}{cc}e^{\cos x} \sin x, & \text { for }|x| \leq 2 \ 2 & \text { otherwise }\end{array}$ then $\int _{-2}^{3} f(x) d x$ is equal to

$(2000,2 M)$

(a) 0

(b) 1

(c) 2

(d) 3

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Solution:

  1. Given, $f(x)=2$, otherwise

$\therefore \int _{-2}^{3} f(x) d x=\int _{-2}^{2} f(x) d x+\int _2^{3} f(x) d x$

$$ =\int _{-2}^{2} e^{\cos x} \sin x d x+\int _2^{3} 2 d x=0+2[x] _2^{3} $$

$\left[\because e^{\cos x} \sin x\right.$ is an odd function]

$$ =2[3-2]=2 \quad \because \int _{-2}^{3} f(x) d x=2 $$



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