Definite Integral to Infinity of Cube of Sine x over x Cubed
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Theorem
- $\ds \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$
Proof
Let:
- $\ds \map I \alpha = \int_0^\infty \frac {\map {\sin^3} {\alpha x} } {x^3} \rd x$
for positive real parameter $\alpha$.
We have:
\(\ds \map I 0\) | \(=\) | \(\ds \int_0^\infty \frac {\map {\sin^3} {0 x} } {x^3} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \frac 0 {x^3} \rd x\) | Sine of Zero is Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
We aim to evaluate explicitly:
- $\ds \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \map I 1$
Differentiating with respect to $\alpha$ we have:
\(\ds \map {I'} \alpha\) | \(=\) | \(\ds \frac \d {\d \alpha} \int_0^\infty \frac {\map {\sin^3} {\alpha x} } {x^3} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \frac \partial {\partial \alpha} \paren {\frac {\map {\sin^3} {\alpha x} } {x^3} } \rd x\) | Definite Integral of Partial Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \int_0^\infty \frac \partial {\partial \alpha} \paren {\frac {3 \map \sin {\alpha x} - \map \sin {3 \alpha x} } {x^3} } \rd x\) | Cube of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 4 \int_0^\infty \frac {\map \cos {\alpha x} - \map \cos {3 \alpha x} } {x^2} \rd x\) | Derivative of Cosine Function, Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 4 \times \frac \pi 2 \paren {3 \alpha - \alpha}\) | Definite Integral to Infinity of $\dfrac {\cos p x - \cos q x} {x^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \alpha \pi} 4\) |
We therefore have:
\(\ds \int_0^1 \map {I'} \alpha \rd \alpha\) | \(=\) | \(\ds \intlimits {\frac {3 \alpha^2 \pi} 8} 0 1\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \pi} 8\) |
By Fundamental Theorem of Calculus: Second Part, we also have:
- $\ds \int_0^1 \map {I'} \alpha \rd \alpha = \map I 1 - \map I 0 = \map I 1$
giving:
- $\ds \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.58$