SATHEE: Indefinite Integration 1 Question 11

Indefinite Integration 1 Question 11

12. The value of the integral $\int_{- 2}^{2} \frac{\sin^{2} x}{\frac{x}{π} + \frac{1}{2}} d x$

(where, $[x]$ denotes the greatest integer less than or equal to $x$ ) is

(2019 Main, 11 Jan I)

(a) $4 - \sin 4$

(b) 4

(d) 0

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Solution:

Let $I = \int_{- 2}^{2} \frac{\sin^{2} x}{\frac{1}{2} + \frac{x}{π}} d x$

Also, let $f (x) = \frac{\sin^{2} x}{\frac{1}{2} + \frac{x}{π}}$

Then, $f (- x) = \frac{\sin^{2} (- x)}{\frac{1}{2} + - \frac{x}{π}} ($ replacing $x$ by $- x)$

$\begin{aligned} = \frac{\sin^{2} x}{\frac{1}{2} + - 1 - \frac{x}{π}} ∵ [- x] = \begin{array}{cc} - [x], & if x \in I \\ - 1 - [x], & if x \notin I \end{array} \\ \Rightarrow f (- x) & = - \frac{\sin^{2} x}{\frac{1}{2} + \frac{x}{π}} = - f (x) \end{aligned}$

i.e. $f (x)$ is odd function

$∴ I = 0 ∵ \int_{- a}^{a} f (x) d x = \begin{matrix} 0, if f (x) is odd function \\ 2 \int_{0}^{a} f (x) d x, if f (x) is even function \end{matrix}$

Indefinite Integration 1 Question 11

12. The value of the integral $\int_{- 2}^{2} \frac{\sin^{2} x}{\frac{x}{π} + \frac{1}{2}} d x$

Solution:

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Indefinite Integration 1 Question 11

12. The value of the integral ∫−22sin2⁡xxπ+12dx

Solution:

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12. The value of the integral $\int_{- 2}^{2} \frac{\sin^{2} x}{\frac{x}{π} + \frac{1}{2}} d x$