1. **Problem statement:** Find the Fourier cosine integral representation of the function $$f(x) = \begin{cases} \sin x, & 0 \leq x \leq \pi \\ 0, & x > \pi \end{cases}$$ 2. **Fourier cosine integral formula:** The Fourier cosine integral representation of a function is given by $$f(x) = \int_0^\infty A(\omega) \cos(\omega x) \, d\omega$$ where $$A(\omega) = \frac{2}{\pi} \int_0^\infty f(t) \cos(\omega t) \, dt$$ 3. **Apply the formula to our function:** Since $f(t) = \sin t$ for $0 \leq t \leq \pi$ and $0$ otherwise, the integral for $A(\omega)$ becomes $$A(\omega) = \frac{2}{\pi} \int_0^\pi \sin t \cos(\omega t) \, dt$$ 4. **Evaluate the integral for $A(\omega)$:** Use the product-to-sum identity: $$\sin t \cos(\omega t) = \frac{1}{2} [\sin(t + \omega t) + \sin(t - \omega t)] = \frac{1}{2} [\sin((1+\omega)t) + \sin((1-\omega)t)]$$ So, $$A(\omega) = \frac{2}{\pi} \cdot \frac{1}{2} \int_0^\pi [\sin((1+\omega)t) + \sin((1-\omega)t)] \, dt = \frac{1}{\pi} \int_0^\pi [\sin((1+\omega)t) + \sin((1-\omega)t)] \, dt$$ 5. **Integrate each sine term:** $$\int_0^\pi \sin(a t) \, dt = \left[-\frac{\cos(a t)}{a}\right]_0^\pi = \frac{1 - \cos(a \pi)}{a}$$ Apply this for $a = 1+\omega$ and $a = 1-\omega$: $$\int_0^\pi \sin((1+\omega)t) \, dt = \frac{1 - \cos((1+\omega)\pi)}{1+\omega}$$ $$\int_0^\pi \sin((1-\omega)t) \, dt = \frac{1 - \cos((1-\omega)\pi)}{1-\omega}$$ 6. **Combine results:** $$A(\omega) = \frac{1}{\pi} \left[ \frac{1 - \cos((1+\omega)\pi)}{1+\omega} + \frac{1 - \cos((1-\omega)\pi)}{1-\omega} \right]$$ 7. **Simplify cosine terms:** Since $\cos(\alpha \pi) = (-1)^\alpha$ for integer $\alpha$, and $\omega$ is continuous, keep the expression as is for general $\omega$. 8. **Final Fourier cosine integral representation:** $$f(x) = \int_0^\infty A(\omega) \cos(\omega x) \, d\omega$$ with $$A(\omega) = \frac{1}{\pi} \left[ \frac{1 - \cos((1+\omega)\pi)}{1+\omega} + \frac{1 - \cos((1-\omega)\pi)}{1-\omega} \right]$$ This integral representation reconstructs $f(x)$ from its cosine transform coefficients $A(\omega)$. **Answer:** $$\boxed{f(x) = \int_0^\infty \frac{1}{\pi} \left[ \frac{1 - \cos((1+\omega)\pi)}{1+\omega} + \frac{1 - \cos((1-\omega)\pi)}{1-\omega} \right] \cos(\omega x) \, d\omega}$$