Should we really integrate?

Calculus Level 4

∫ − ∞ ∞ sin ⁡ 4 x 2 x d x = ? \large \displaystyle \int_{-\infty}^{\infty} \dfrac{\sin 4x}{2x} \, dx = \, ?

2 π 2\pi π \pi π 2 \frac{\pi}{2} 4 π 4\pi

Abhay Tiwari by Abhay Tiwari , 28, India

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3 solutions

Chew-Seong Cheong
Apr 22, 2016

I = ∫ − ∞ ∞ sin ⁡ ( 4 x ) 2 x d x As sin ⁡ ( 4 x ) 2 x is even = ∫ 0 ∞ sin ⁡ ( 4 x ) x d x Let u = 4 x ⟹ d u = 4 d x = ∫ 0 ∞ sin ⁡ u u d u Dirichlet integral = π 2 \begin{aligned} I & = \int_{-\infty}^\infty \color{#3D99F6}{\frac{\sin (4x)}{2x}}\, dx \quad \quad \small \color{#3D99F6}{\text{As }\frac{\sin (4x)}{2x} \text{ is even}} \\ & = \int_{0}^\infty \frac{\sin (\color{#3D99F6}{4x})}{x} \, dx \quad \quad \small \color{#3D99F6}{\text{Let }u = 4x \implies du = 4\, dx} \\ & = \color{#3D99F6}{\int_{0}^\infty \frac{\sin u}{u} \, du} \quad \quad \small \color{#3D99F6}{\text{Dirichlet integral}} \\ & = \boxed{\color{#3D99F6}{\dfrac{\pi}{2}}} \end{aligned}

About Dirichlet integral .

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Thanks a lot sir, I was wondering if someone would post this. Besides my solution , This is also a good approach. But sir can you solve this through the above integral. It would be really appreciated. Thanks a lot again.

Abhay Tiwari - 5 years, 1 month ago

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I have just done the solution. Please check.

Chew-Seong Cheong - 5 years, 1 month ago
Abhay Tiwari
Apr 22, 2016

Did it through DUALITY property of Fourier Transform.

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Hobart Pao
Apr 23, 2016

With knowledge of the Dirichlet integral, this can be computed mentally!

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