To solve the integral equation:

$\int_1^{a}\frac{\sqrt{\pi}}{\pi x}\, dx = 0.8$

Let's proceed with the steps:

Step 1: Simplify the Integral Expression

The given integral is:

$\int_1^{a}\frac{\sqrt{\pi}}{\pi x}\, dx$

This can be simplified by factoring out the constants:

$\frac{\sqrt{\pi}}{\pi} \int_1^{a} \frac{1}{x}\, dx$

Step 2: Integrate the Function

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln|x|$. Therefore:

$\frac{\sqrt{\pi}}{\pi} \left[ \ln|x| \right]_1^a$

Substituting the limits:

$\frac{\sqrt{\pi}}{\pi} \left( \ln a - \ln 1 \right)$

Since $\ln 1 = 0$, this simplifies to:

$\frac{\sqrt{\pi}}{\pi} \ln a$

Step 3: Set the Integral Equal to 0.8

Now set the integral equal to 0.8:

$\frac{\sqrt{\pi}}{\pi} \ln a = 0.8$

Step 4: Solve for $a$

To solve for $a$, multiply both sides by $\frac{\pi}{\sqrt{\pi}}$:

$\ln a = \frac{0.8\pi}{\sqrt{\pi}}$

Finally, exponentiate both sides to solve for $a$:

$a = e^{\frac{0.8\pi}{\sqrt{\pi}}}$

This is the value of $a$ that satisfies the given integral equation.

Final Expression for $a$:

$a = e^{0.8\sqrt{\pi}}$

This is the required value of $a$.

Would you like any further explanation or details on this solution?

Related Questions:

1. What are the properties of logarithmic integrals?
2. How does changing the limits of integration affect the result?
3. What is the significance of the constant $e$ in solving integrals?
4. How does the exponential function relate to logarithmic integrals?
5. Can we solve similar integrals using substitution methods?
6. What are common applications of logarithmic integrals in physics?
7. How do you interpret the result $a = e^{0.8\sqrt{\pi}}$ geometrically?
8. What methods can be used to approximate the value of $a$?

Tip: When dealing with logarithmic integrals, always ensure that the argument inside the logarithm is positive to avoid undefined results.