Example 3. Solve the integral equation:

$$
\int_{0}^{+\infty} \varphi(t) \sin x t \, d t=e^{-x} \quad(x>0)
$$
Solution. The function $\sqrt{\frac{2}{\pi}} \mathrm{e}^{-z}$ is obviously the sine Fourier transform of the desired function $\varphi(t)$. Applying the inversion formula (6) of the sine Fourier transform, we will have

$$
\varphi(t)=\sqrt{\frac{2}{\pi}} \int_{0}^{+\infty} \sqrt{\frac{2}{\pi}} e^{-x} \sin x t d x=\frac{2}{\pi} \int_{0}^{+\infty} e^{-x} \sin x t d x
$$

The integral on the right side (7) is calculated using integration by parts twice. We will get

$$
\int_{0}^{+\infty} e^{-x} \sin t x d x=\frac{t}{1+t^{2}}
$$

so that

$$
\varphi(t)=\frac{2}{\pi} \frac{t}{1+t^{2}}
$$