Higher powers

Calculus Level 5

E = ∫ − π / 2 0 cos ⁡ 21 ( x ) sin ⁡ 17 ( x ) d x \large \mathscr{E} = \displaystyle \int_{-\pi/2}^{0} \cos^{21}(x) \sin^{17}(x) \, dx

If E \mathscr{E} can be expressed in the form − a b - \dfrac{a}{b} , with a a and b b are coprime positive integers , find a + b a + b .

No calculators allowed.


The answer is 1662805.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

Chew-Seong Cheong
Jun 16, 2016

E = ∫ − π 2 0 cos ⁡ 21 x sin ⁡ 17 x d x Let u = − x ⟹ cos ⁡ x = cos ⁡ u , sin ⁡ x = − sin ⁡ u , d u = − d x = ∫ π 2 0 cos ⁡ 21 u sin ⁡ 17 u d u = − ∫ 0 π 2 cos ⁡ 21 u sin ⁡ 17 u d u = − 1 2 B ( 11 , 9 ) B ( m , n ) is beta function = − 1 2 ⋅ Γ ( 11 ) Γ ( 9 ) Γ ( 20 ) Γ ( n ) is gamma function = − 1 2 ⋅ 10 ! 8 ! 19 ! = − 1 1662804 \begin{aligned} \mathscr E & = \int_{-\frac \pi 2}^0 \cos^{21} x \ \sin^{17} x \ dx \quad \quad \small \color{#3D99F6}{\text{Let }u = - x \implies \cos x = \cos u, \ \sin x = - \sin u, \ du = - dx} \\ & = \int_{\frac \pi 2}^0 \cos^{21} u \ \sin^{17} u \ du \\ & = - \int ^{\frac \pi 2}_0 \cos^{21} u \ \sin^{17} u \ du \\ & = - \frac 12 \color{#3D99F6}{B \left(11,9 \right) \quad \quad \small B(m,n) \text{ is beta function}} \\ & = - \frac 12 \cdot \color{#3D99F6}{\frac{\Gamma (11) \Gamma (9)}{\Gamma(20)} \quad \quad \small \Gamma (n) \text{ is gamma function}} \\ & = - \frac 12 \cdot \color{#3D99F6}{\frac{10! 8!}{19!}} \\ & = - \frac 1{1662804} \end{aligned}

⟹ a + b = 1 + 1662804 = 1662805 \implies a + b = 1+1662804 = \boxed{1662805}

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