Not my calculus problem #1

Calculus Level 2

∫ 0 π ( 3 x 2 − 4 ) cos ⁡ x d x = a π \int_0^\pi (3x^2 - 4) \cos x \ dx = a \pi

Find the real value a a satisfying the equation above.

Shared by Basel Almalete


The answer is -6.

Chew-Seong Cheong by Chew-Seong Cheong , 63, Malaysia

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1 solution

Chew-Seong Cheong
Jul 24, 2020

The integral can be solved using integration by parts .

I = ∫ 0 π ( 3 x 2 − 4 ) cos ⁡ x d x = ( 3 x 2 − 4 ) sin ⁡ x ∣ 0 π − ∫ 0 π 6 x sin ⁡ x d x = 0 + 6 x cos ⁡ x ∣ 0 π − ∫ 0 π 6 cos ⁡ x d x = − 6 π − 6 sin ⁡ x ∣ 0 π = − 6 π \begin{aligned} I & = \int_0^\pi (3x^2-4) \cos x \ dx \\ & = (3x^2-4)\sin x \ \bigg|_0^\pi - \int_0^\pi 6x \sin x \ dx \\ & = 0 + 6 x \cos x \ \bigg|_0^\pi - \int_0^\pi 6 \cos x \ dx \\ & = - 6 \pi - 6 \sin x \ \bigg|_0^\pi \\ & = - 6 \pi \end{aligned}

Therefore a = − 6 a = \boxed {-6} .

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