Evaluate ∫ R sin 4 ⁡ ( x ) x 2 d x

If we use integration by parts and compute a Fourier sine series we have: ∫ 0 + ∞ sin 4 ⁡ ( x ) x 2 d x = ∫ 0 + ∞ 4 cos ⁡ ( x ) sin 3 ⁡ ( x ) x d x = ∫ 0 + ∞ sin ⁡ ( 2 x ) − 1 2 sin ⁡ ( 4 x ) x = π 4 since for every n ∈ N + ∫ 0 + ∞ sin ⁡ ( n x ) n d x = ∫ 0 + ∞ sin ⁡ z z d z = π 2 .

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logiski9s

Answered question

2022-07-09

Evaluate

\int_{R} \frac{\sin^{4} (x)}{x^{2}} d x

Answer & Explanation

tilsjaskak6

Beginner2022-07-10Added 14 answers

If we use integration by parts and compute a Fourier sine series we have:

\int_{0}^{+ \infty} \frac{\sin^{4} (x)}{x^{2}} d x = \int_{0}^{+ \infty} \frac{4 \cos (x) \sin^{3} (x)}{x} d x = \int_{0}^{+ \infty} \frac{\sin (2 x) - \frac{1}{2} \sin (4 x)}{x} = \frac{π}{4}

since for every

n \in N^{+}

\int_{0}^{+ \infty} \frac{\sin (n x)}{n} d x = \int_{0}^{+ \infty} \frac{\sin z}{z} d z = \frac{π}{2} .

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