Definite Integration Ques 5
- The value of the integral $\int _{-2}^{2} \frac{\sin ^{2} x}{\frac{x}{\pi}+\frac{1}{2}} d x$
(where, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ ) is
(2019 Main, 11 Jan I)
(a) $4-\sin 4$
4
(c) $\sin 4$
0
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Solution:
- Let $I=\int _{-2}^{2} \frac{\sin ^{2} x}{\frac{1}{2}+\frac{x}{\pi}} d x$
Also, let $f(x)=\frac{\sin ^{2} x}{\frac{1}{2}+\frac{x}{\pi}}$
Then, $f(-x)=\frac{\sin ^{2}(-x)}{\frac{1}{2}-\frac{x}{\pi}}$ (replacing $x$ by $-x)$
$$ \begin{aligned} & =\frac{\sin ^{2} x}{\frac{1}{2}-1-\frac{x}{\pi}} \because[-x]=\begin{array}{cc} -[x], & \text { if } x \in I \text{ and } x \notin \partial I \\ -1-[x], & \text { if } x \notin I \end{array} \ \Rightarrow \quad f(-x) & =-\frac{\sin ^{2} x}{\frac{1}{2}+\frac{x}{\pi}}=-f(x) \end{aligned} $$
i.e. $f(x)$ is odd function
$$ \therefore I=0 \quad \because \int _{-a}^{\infty} f(x) d x=\begin{gathered} 0, \text { if } f(x) \text { is an odd function } \\ 2 \int _0^{a} f(x) d x, \text { if } f(x) \text { is even function } \end{gathered} $$
BETA
