Integration 0 → π 2 0\rightarrow\dfrac{\pi}{2}

Calculus Level 3

∫ 0 π 2 x 2 cos ⁡ x d x = π a b − c \large \int_{0}^{\frac{\pi}{2}}{x^2\cos x} \ dx=\dfrac{\pi^a}{b}-c

Positive integers a a , b b and c c satisfy the above equation. Find the value of a + b + c a+b+c .


The answer is 8.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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1 solution

Chew-Seong Cheong
Dec 11, 2017

I = i n t 0 π 2 x 2 cos ⁡ x d x Using integration by parts = x 2 sin ⁡ x + 2 x cos ⁡ x − 2 sin ⁡ x ∣ 0 π 2 = π 2 4 + 0 − 2 \begin{aligned} I & = int_0^\frac \pi 2 x^2 \cos x \ dx & \small \color{#3D99F6} \text{Using integration by parts} \\ & = x^2\sin x + 2x\cos x - 2\sin x \bigg|_0^\frac \pi 2 \\ & = \frac {\pi^2}4 + 0 - 2 \end{aligned}

⟹ a + b + c = 2 + 4 + 2 = 8 \implies a+b+c = 2+4+2 = \boxed{8}

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