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Calculus Level 3

∫ 0 π e cos ⁡ 2 x cos ⁡ 3 ( 2 n + 1 ) x d x \large \int_0^\pi e^{\cos^{2} x} \cos^{3} (2n+1)x \ dx

What is the value of the integral above, where n n is an Integer?


The answer is 0.

Parth Lohomi by Parth Lohomi , 20, India

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2 solutions

Parth Lohomi
Dec 1, 2014

Let f(x)= e c o s 2 x e^{cos^{2}x} c o s 3 ( 2 n + 1 ) x cos^{3}(2n+1)x

Then

f( π \pi -x)= e c o s 2 π − x e^{cos^{2}\pi-x} c o s 3 ( 2 n + 1 ) π − x cos^{3}(2n+1)\pi-x

= e c o s 2 x e^{cos^{2}x} c o s 3 ( 2 n π + π − ( 2 n + 1 ) x cos^{3}(2n\pi+\pi-(2n+1)x = e c o s 2 x e^{cos^{2}x} c o s 3 π − ( 2 n + 1 ) x cos^{3}\pi-(2n+1)x

= − - e c o s 2 x e^{cos^{2}x} c o s 3 ( 2 n + 1 ) x cos^{3}(2n+1)x = − f ( x ) -f(x)

Therefore The integral = 0 0

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Or observe that for all integer values of n n , since cos ⁡ 3 ( 2 n + 1 ) \cos^3(2n+1) is non-constant and never equal to 0 0 and a definite integral is constant, the integral must be equal to 0 0 as a result.

Jake Lai - 6 years, 6 months ago

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Yup that's good!!

Parth Lohomi - 6 years, 6 months ago

Did the same. By the way problems of this set is really nice. I would like to do some more like these.

Trishit Chandra - 6 years, 3 months ago
Chew-Seong Cheong
Aug 30, 2018

I = ∫ 0 π e cos ⁡ 2 x cos ⁡ 3 ( ( 2 n + 1 ) x ) d x By ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = ∫ 0 π ( e cos ⁡ 2 x cos ⁡ 3 ( ( 2 n + 1 ) x ) + e cos ⁡ 2 ( π − x ) cos ⁡ 3 ( ( 2 n + 1 ) ( π − x ) ) ) d x Note that cos ⁡ ( π − θ ) = − cos ⁡ θ = ∫ 0 π ( e cos ⁡ 2 x cos ⁡ 3 ( ( 2 n + 1 ) x ) + e cos ⁡ 2 x cos ⁡ 3 ( π − ( 2 n + 1 ) x ) ) d x = ∫ 0 π ( e cos ⁡ 2 x cos ⁡ 3 ( ( 2 n + 1 ) x ) − e cos ⁡ 2 x cos ⁡ 3 ( ( 2 n + 1 ) x ) ) d x = 0 \begin{aligned} I & = \int_0^\pi e^{\cos^2 x} \cos^3 ((2n+1) x) \ dx & \small \color{#3D99F6} \text{By }\int_a^b f(x) \ dx = \int_a^b f(a+b - x) \ dx \\ & = \int_0^\pi \left(e^{\cos^2 x} \cos^3 ((2n+1) x) + e^{\color{#3D99F6}\cos^2 (\pi - x)} \cos^3 ((2n+1)(\pi- x)) \right) dx & \small \color{#3D99F6} \text{Note that }\cos (\pi - \theta) = - \cos \theta \\ & = \int_0^\pi \left(e^{\cos^2 x} \cos^3 ((2n+1) x)\ {\color{#D61F06}+}\ e^{\color{#3D99F6}\cos^2 x} {\color{#D61F06}\cos^3 (\pi-(2n+1)x)} \right) dx \\ & = \int_0^\pi \left(e^{\cos^2 x} \cos^3 ((2n+1) x)\ {\color{#D61F06}-}\ e^{\cos^2 x} {\color{#D61F06}\cos^3 ((2n+1)x)} \right) dx \\ & = \boxed 0 \end{aligned}

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