Question

Evaluate the integral: \[ \int_0^1 \left( \sqrt{10} \right)^{2x} \, dx \]

• A. \( \frac{10}{\log 10} \)
• B. \( \frac{9}{\log 10} \)
• C. \( \frac{1}{\log 10} \)
• D. \( \frac{4}{\log 5} \)

Hint:

When encountering exponential integrals with a base other than \( e \), use logarithms to simplify the integral.

Answer 1

The Correct Option is B

We are given the integral: \[ I = \int_0^1 \left( \sqrt{10} \right)^{2x} \, dx = \int_0^1 10^x \, dx \] Step 1: Rewrite the integrand as \( 10^x \). Step 2: The integral of \( 10^x \) is: \[ \int 10^x \, dx = \frac{10^x}{\log 10} \] Step 3: Evaluate the integral from 0 to 1: \[ I = \left[ \frac{10^x}{\log 10} \right]_0^1 = \frac{10^1 - 10^0}{\log 10} = \frac{10 - 1}{\log 10} = \frac{9}{\log 10} \] Thus, the correct answer is \( \frac{9}{\log 10} \). % Final Answer The value of the integral is \( \frac{9}{\log 10} \).