If ∫1a(3x2+2x+1) dx = 11, find the real value of a - Mathematics and Statistics

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Sum

If `int_1^"a" (3x^2 + 2x + 1)  "d"x` = 11, find the real value of a

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Solution

Given, `int_1^"a" (3x^2 + 2x + 1)  "d"x` = 11

∴ `[(3x^3)/3 + (2x^2)/2 + x]_1^"a"` = 11

∴ `[x^3 + x^2 + x]_1^"a"` = 11

∴ (a3 + a2 + a) – (1 + 1 + 1) = 11

∴ a3 + a2 + a – 3 = 11

∴ a3 + a2 + a – 14 = 0 

∴ (a – 2)(a2 + 3a + 7) = 0

∴ a = 2 or a2 + 3a + 7 = 0

But, a2 + 3a + 7 = 0 does not have real roots.

∴ a = 2

Concept: Fundamental Theorem of Integral Calculus
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Chapter 1.6: Definite Integration - Q.4

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