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JEE Main 202229 JuneMorning Shift
Exam
JEE Main
Year
2022
Date
29 June
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2022 · 29 June · Morning Shift · Mathematics · Definite Integration

∫ _ 0 ^ 5 (π (x - [(x)/(2)]))dx , where [t] denotes greatest...

{∫}_{0}^{5}\cos \left(\pi\left(x - \left[\dfrac{x}{2}\right]\right)\right)dx,where[t], where \left[t\right] denotes greatest integer less than or equal to tt, is equal to

Options

  1. A00
  2. B22
  3. C−3- 3
  4. D44

Correct answer

A. 00

Step-by-step solution

I=∫05cos⁡([x2]π−\pix)dxI = {∫}_{0}^{5}\cos \left(\left[\dfrac{x}{2}\right]\pi - \pix\right)dx

∵x∈(0,5)∴x∈(0,52)⇒[x2]∈{0,1,2}∵ x \in \left(0, 5\right) ∴ x \in \left(0, \dfrac{5}{2}\right) ⇒ \left[\dfrac{x}{2}\right] \in \left\{0, 1, 2\right\}

I=∫02cos⁡(−\pix)dx+∫24cos⁡(π−\pix)dx+∫45cos⁡(2π−\pix)dxI = {∫}_{0}^{2}\cos \left(- \pix\right)dx + {∫}_{2}^{4}\cos \left(\pi - \pix\right)dx + {∫}_{4}^{5}\cos \left(2\pi - \pix\right)dx

⇒I=[sin\pixπ]02+[−sin\pixπ]24+[sin\pixπ]45=0⇒ I = {\left[\dfrac{sin \pix}{\pi}\right]}_{0}^{2} + {\left[- \dfrac{sin \pix}{\pi}\right]}_{2}^{4} + {\left[\dfrac{sin \pix}{\pi}\right]}_{4}^{5} = 0

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