Improper Trigonometric Integrals

Integrals of a rational function multiplied by $\cos(\alpha x)$ or $\sin(\alpha x)$ use

and compare real and imaginary parts.

Existence

Let $R(z) = \frac{P(z)}{Q(z)}$ where $P$ and $Q$ are polynomials with real coefficients of degrees $m$ and $n$. Suppose

• $Q(x) \neq 0$ for all $x \in \mathbb{R}$
• $n \ge m + 1$

Then for each $\alpha > 0$, the Cauchy principal values

exist.

Evaluation

Under the same conditions with $\alpha > 0$, let

If $z_1, z_2, \ldots, z_k$ are the poles of $F$ in the upper half-plane, then

Consequently,

Examples:

• $\text{P.V.} \int_{-\infty}^{\infty} \frac{x\sin x}{x^2+4}\,\text{d}x = \frac{\pi}{e^2}$
• $\text{P.V.} \int_{-\infty}^{\infty} \frac{\cos x}{x^4+4}\,\text{d}x = \frac{\pi(\cos 1 + \sin 1)}{4e}$

Worked Examples

Cosine Integral

Evaluate $\text{P.V.} \int_{-\infty}^{\infty} \frac{\cos x}{x^2+1}\,\text{d}x$.

Here $m = 0$, $n = 2 \ge m + 1$, $\alpha = 1$, and $F(z) = \frac{e^{iz}}{z^2+1}$. The only pole in the upper half-plane is $z = i$.

Then $\text{Res}(F, i) = -\frac{i}{2e}$, so $\text{Im}\left(\text{Res}(F, i)\right) = -\frac{1}{2e}$ and $\text{Re}\left(\text{Res}(F, i)\right) = 0$.

Sine Integral

Show that $\text{P.V.} \int_{-\infty}^{\infty} \frac{x \sin x}{x^2+4}\,\text{d}x = \frac{\pi}{e^2}$.

Here $m = 1$, $n = 2 \ge m + 1$, $\alpha = 1$, and $F(z) = \frac{z e^{iz}}{z^2+4}$. The only pole in the upper half-plane is $z = 2i$.

Two Poles

Show that $\text{P.V.} \int_{-\infty}^{\infty} \frac{\cos x}{x^4+4}\,\text{d}x = \frac{\pi(\cos 1 + \sin 1)}{4e}$.

Here $F(z) = \frac{e^{iz}}{z^4+4}$. The roots of $z^4 = -4$ in the upper half-plane are $z = 1 + i$ and $z = -1 + i$.

For a simple pole $z_k$, using $z_k^4 = -4$,

For $z = 1 + i$, $e^{iz} = e^{-1} e^{i}$.

For $z = -1 + i$, $e^{iz} = e^{-1} e^{-i}$.

Expand the products.

The sum of the residues is