Example 8. Calculate the integral

$$
I=\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)^{2}} \quad(a>0)
$$
Solution. Since the integrand $f(x)=\frac{x^{2}}{\left(x^{2}+a^{2}\right)^{2}}$ is an even function,

$$
I=\frac{1}{2} \int_{-\infty}^{+\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)^{2}}
$$

Introduce the function $f(z)=\frac{z^{2}}{\left(z^{2}+a^{2}\right)^{2}}$; which on the real axis, i.e., when $z=x$, coincides with $f(x)$. The function $f(z)$ has a pole of second order in the upper half-plane at the point $z=a i$. The residue of $f(z)$ at this pole is

$$
\operatorname{res} f(a i)=\lim _{z \rightarrow a i} \frac{d}{d z}\left[f(z)(z-a i)^{2}\right]=\lim _{z \rightarrow a i} \frac{d}{d z}\left[\frac{z^{2}}{(z+a i)^{2}}\right]=\lim _{z \rightarrow a i} \frac{2 a i z}{(z+a i)^{3}}=\frac{1}{4 a i}
$$

Using formula (4), we get

$$
I=\frac{1}{2} \int_{-\infty}^{+\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)^{2}}=\frac{1}{2} \cdot 2 \pi i \cdot \frac{1}{4 a i}=\frac{\pi}{4 a}
$$

## Problems for Independent Solution

Compute the following integrals with infinite limits:

379. $\int_{0}^{\infty} \frac{x^{2}+1}{x^{4}+1} d x$.
380. $\int_{-\infty}^{+\infty} \frac{d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}(a>0, b>0)$.
381. $\int_{-\infty}^{+\infty} \frac{d x}{\left(x^{2}+1\right)^{3}}$.
382. $\int_{-\infty}^{+\infty} \frac{d x}{\left(1+x^{2}\right)^{n+1}}$.
383. $\int_{-\infty}^{+\infty} \frac{x d x}{\left(x^{2}+4 x+13\right)^{2}}$.
384. $\int_{-\infty}^{+\infty} \frac{d x}{\left(x^{2}+a^{2}\right)^{2}\left(x^{2}+b^{2}\right)^{2}}$ 385. $\int_{0}^{\infty} \frac{x+1}{x^{6}+1} d x$.
385. $\int_{-\infty}^{+\infty} \frac{x^{2 m}}{1+x^{2 n}} d x \quad$ 387. $\int_{-\infty}^{+\infty} \frac{d x}{1+x^{6}} \quad$ 388. $\int_{-\infty}^{+\infty} \frac{d x}{\left(x^{2}+2 x+2\right)^{2}}$.
386. $\int_{-\infty}^{+\infty} \frac{x^{4} d x}{\left(a+b x^{2}\right)^{4}}(a>0, b>0)$.
387. Prove the formula $\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{n+1}}=\frac{1 \cdot 3 \ldots(2 n-1)}{2 \cdot 4 \cdot 6 \ldots 2 n} \pi$.

## 2. Integrals of the Form

$$
\int_{0}^{\infty} R(x) \cos \lambda x d x, \quad \int_{0}^{\infty} R(x) \sin \lambda x d x
$$

where $R(x)$ is a proper rational function, $\lambda>0$ is any real number.

When computing such integrals, it is convenient to use the following lemma:

Jordan's Lemma. Let $g(z)$ be a function that is analytic in the upper half-plane ($00$)

$$
\lim _{R \rightarrow \infty} \int_{C_{R}} g(z) e^{i \lambda z} d z=0
$$

where the contour $C_{R}$ is a semicircle in the upper half-plane with center at the point 0 and radius $R$ (Fig. 7).

![](https://cdn.mathpix.com/cropped/2024_05_22_f7d63c3a5e94c3f2f1bbg-095.jpg?height=323&width=463&top_left_y=998&top_left_x=816)

Fig. 7.