1. **Stating the problem:** We are given the integral equation: $$\frac{1}{a} \pi \int_{8}^{1.\pi} \frac{d}{a + b \cos t + 5} = \frac{8}{\sqrt{a^2 - b^2}}$$ 2. **Understanding the integral:** This looks like a form of the integral of the type: $$\int \frac{dt}{a + b \cos t}$$ which has a known solution: $$\int_0^{2\pi} \frac{dt}{a + b \cos t} = \frac{2\pi}{\sqrt{a^2 - b^2}}$$ provided $|a| > |b|$. 3. **Analyzing the given integral:** - The integral limits are from 8 to $1.\pi$ (which likely means $\pi$). - The numerator has a constant $d$. - There is a factor $\frac{1}{a} \pi$ outside the integral. 4. **Simplifying the integral expression:** Assuming the integral is: $$\int_8^{\pi} \frac{d}{a + b \cos t + 5} dt$$ We can factor constants: $$d \int_8^{\pi} \frac{dt}{a + b \cos t + 5}$$ 5. **Relating to the known formula:** If we let $A = a + 5$, then the integral becomes: $$d \int_8^{\pi} \frac{dt}{A + b \cos t}$$ 6. **Final equation:** Given the equation: $$\frac{1}{a} \pi d \int_8^{\pi} \frac{dt}{a + b \cos t + 5} = \frac{8}{\sqrt{a^2 - b^2}}$$ 7. **Summary:** The problem involves evaluating or verifying this integral identity involving trigonometric integrals and constants $a$, $b$, and $d$. Since the problem statement is transcription and no specific question is asked, this is the interpretation and explanation of the integral given.