Integral to Infinity of Sine p x Sine q x over x Squared
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Theorem
- $\ds \int_0^\infty \frac {\sin p x \sin q x} {x^2} \rd x = \begin {cases} \dfrac {\pi p} 2 & : 0 < p \le q \\ \dfrac {\pi q} 2 & : p \ge q > 0 \end {cases}$
Proof
With a view to expressing the primitive in the form:
- $\ds \int f g' \rd t = f g - \int f' g \rd t$
let:
\(\ds f\) | \(=\) | \(\ds \sin p x \sin q x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds f'\) | \(=\) | \(\ds p \cos p x \sin q x + q \sin p x \cos q x\) | Product Rule for Derivatives and Derivative of $\sin a x$ | ||||||||||
\(\ds g'\) | \(=\) | \(\ds \frac 1 {x^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g\) | \(=\) | \(\ds -\frac 1 x\) | Chain Rule for Derivatives |
So:
\(\ds \int_0^\infty \frac {\sin p x \sin q x} {x^2} \rd x\) | \(=\) | \(\ds \intlimits {-\frac {\sin p x \sin q x} x} 0 \infty + \int_0^\infty \frac {p \cos p x \sin q x + q \sin p x \cos q x} x \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + \int_0^\infty \frac {p \cos p x \sin q x + q \sin p x \cos q x} x \rd x\) | evaluating limits using L'Hôpital's Rule and Product Rule for Limits of Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds p \cdot \int_0^\infty \frac {\cos p x \sin q x} x \rd x + q \cdot \int_0^\infty \frac {\sin p x \cos q x} x \rd x\) | Linear Combination of Definite Integrals |
Case $0 < p = q$
\(\ds \) | \(\) | \(\ds p \cdot \int_0^\infty \frac {\cos p x \sin q x} x \rd x + q \cdot \int_0^\infty \frac {\sin p x \cos q x} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 p \cdot \int_0^\infty \frac {\cos p x \sin p x} x \rd x\) | Substitute $p$ for $q$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 p \cdot \frac \pi 4\) | Integral to Infinity of $\dfrac {\sin p x \cos q x} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi p} 2 = \frac {\pi q} 2\) |
$\Box$
Case $0 < p \le q$:
Suppose $0 < p < q$.
Then:
\(\ds \) | \(\) | \(\ds p \cdot \int_0^\infty \frac {\cos p x \sin q x} x \rd x + q \cdot \int_0^\infty \frac {\sin p x \cos q x} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p \cdot \frac \pi 2 + q \cdot 0\) | Integral to Infinity of $\dfrac {\sin p x \cos q x} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi p} 2\) | simplifying |
Adjoin to the case where $p = q$.
$\Box$
Case $p \ge q > 0$:
Suppose $p > q > 0$.
Then:
\(\ds \) | \(\) | \(\ds p \cdot \int_0^\infty \frac {\cos p x \sin q x} x \rd x + q \cdot \int_0^\infty \frac {\sin p x \cos q x} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p \cdot 0 + q \cdot \frac \pi 2\) | Integral to Infinity of $\dfrac {\sin p x \cos q x} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi q} 2\) | simplifying |
Adjoin to the case where $p = q$.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.35$