Let [.] denote the greatest integer function. Then 2 2 ( 12 ( 3 + [ x ] ) 3 + [ sin x ] + [ cos x ] ) ? is equal to: [2026]

Question

Let [.] denote the greatest integer function. Then $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{12 (3 + [x])}{3 + [\sin x] + [\cos x]})$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

$I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{12 (3 + [x])}{3 + [\sin x] + [\cos x]} d x$

$I = \int_{- \frac{π}{2}}^{- 1} \frac{12 (1) d x}{2} + \int_{- 1}^{0} \frac{12 (2) d x}{2} + \int_{0}^{1} \frac{12 (3) d x}{3} + \int_{1}^{\frac{π}{2}} \frac{12 (4) d x}{3}$

$I = 6 (\frac{π}{2} - 1) + 12 (0 + 1) + 12 (1 - 0) + 16 (\frac{π}{2} - 1)$

$I = 3 π - 6 + 12 + 12 + 8 π - 16$

$I = 11 π + 2$

Solution

Q.
Let [.] denote the greatest integer function. Then $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{12 (3 + [x])}{3 + [\sin x] + [\cos x]})$ is equal to: [2026]

1 $13 π + 1$

2 $15 π + 4$

3 $11 π + 2$

4 $12 π + 5$

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Solution

Q. Let [.] denote the greatest integer function. Then ∫-π2π2(12(3+[x])3+[sinx]+[cosx]) is equal to: [2026]

1 13π+1

2 15π+4

3 11π+2

4 12π+5