Question

Evaluate the following definite integrals: if `int_1^"a" (3x^2 + 2x + 1)*  dx` = 11, find a.

Answer 1

Given, `int_1^"a" (3x^2 + 2x + 1)*dx` = 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.