SATHEE: Indefinite Integration 2 Question 1

Indefinite Integration 2 Question 1

1. The value of $\int_{- π / 2}^{π / 2} \frac{d x}{[x] + [\sin x] + 4}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is

(a) $\frac{1}{12} (7 π - 5)$

(b) $\frac{1}{12} (7 π + 5)$

(d) $\frac{3}{20} (4 π - 3)$

(2019 Main, 10 Jan II)

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Answer:

Correct Answer: 1. (d)

Solution:

Let $I = \int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{d x}{[x] + [\sin x] + 4}$

$= \int_{\frac{- π}{2}}^{- 1} \frac{d x}{[x] + [\sin x] + 4} + \int_{- 1}^{0} \frac{d x}{[x] + [\sin x] + 4}$

$+ \int_{0}^{1} \frac{d x}{[x] + [\sin x] + 4} + \int_{1}^{\frac{π}{2}} \frac{d x}{[x] + [\sin x] + 4}$

$∵ [x] = \begin{array}{ll} - 2, & - π / 2 < x < - 1 \\ - 1, & - 1 \leq x < 0 \\ 0, & 0 \leq x < 1 \\ 1, & 1 \leq x < π / 2 \end{array}$

and $[\sin x] = \begin{array}{cc} - 1, & - π / 2 < x < - 1 - 1, & - 1 < x < 0 0, & 0 < x < 1 0, & 1 < x < π / 2 \end{array}$

$[∵$ For $x < 0, - 1 \leq \sin x < 0$ and for $x > 0, 0 < \sin x \leq 1]$

So, $I = \int_{\frac{- π}{2}}^{- 1} \frac{d x}{- 2 - 1 + 4} + \int_{- 1}^{0} \frac{d x}{- 1 - 1 + 4} + \int_{0}^{1} \frac{d x}{0 + 0 + 4}$

$\begin{aligned} = \int_{\frac{π}{2}}^{- 1} \frac{d x}{1} + \int_{- 1}^{0} \frac{d x}{2} + \int_{0}^{1} \frac{d x}{4} + \int_{1}^{\frac{π}{2}} \frac{d x}{5} \\ = - 1 + \frac{π}{2} + \frac{1}{2} (0 + 1) + \frac{1}{4} (1 - 0) + \frac{1}{5} \frac{π}{2} - 1 \\ = - 1 + \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{π}{2} + \frac{π}{10} \\ = \frac{- 20 + 10 + 5 - 4}{20} + \frac{5 π + π}{10} \\ = - \frac{9}{20} + \frac{3 π}{5} = \frac{3}{20} (4 π - 3) \end{aligned}$

Indefinite Integration 2 Question 1

1. The value of $\int_{- π / 2}^{π / 2} \frac{d x}{[x] + [\sin x] + 4}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is

Answer:

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Indefinite Integration 2 Question 1

1. The value of ∫−π/2π/2dx[x]+[sin⁡x]+4, where [t] denotes the greatest integer less than or equal to t, is

Answer:

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1. The value of $\int_{- π / 2}^{π / 2} \frac{d x}{[x] + [\sin x] + 4}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is