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JEE Main202227 Jul 2022Evening ShiftMathematicsDefinite IntegrationActual

Let f x = min x - 1 , x - 2 , … , x - 10 where t denotes the greatest integer ≤ t . Then ∫ 0 10 f x d x + ∫ 0 10 f x 2 d x + ∫ 0 10 f x d x is equal _______. to

Correct answer

0

Step-by-step solution

Given, f x = min x - 1 , x - 2 , … , x - 10 ⇒ f x = x - 10 Now finding ∫ 0 10 f x · d x = - 10 - 9 - 8 - … . . - 1 = - 10 · 11 2 = - 55 Finding ∫ 0 10 f x 2 d x = 10 2 + 9 2 + 8 2 + … + 1 2 = 10 · 11 · 21 6 = 385 And ∫ 0 10 f x = 10 + 9 + 8 + … . + 1 = 10 · 11 2 = 55 So, ∫ 0 10 f x d x + ∫ 0 10 f x 2 d x + ∫ 0 10 f x d x = - 55 + 385 + 55 = 385

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