https://file.aiquickdraw.com/mathbot/file/172553230778592gezvs3.jpg

The integral given in the image is:

\[
\int_0^1 5^x \, dx
\]

To solve this, we need to compute the integral of \( 5^x \) with respect to \( x \).

### Step 1: Integral Formula
We use the formula for integrating an exponential function of the form \( a^x \):

\[
\int a^x \, dx = \frac{a^x}{\ln(a)} + C
\]

where \( a \) is a constant, and \( \ln(a) \) is the natural logarithm of \( a \).

### Step 2: Apply the Formula
In this case, \( a = 5 \), so the integral becomes:

\[
\int 5^x \, dx = \frac{5^x}{\ln(5)}
\]

### Step 3: Apply the Limits of Integration
We need to evaluate this from 0 to 1:

\[
\int_0^1 5^x \, dx = \left[ \frac{5^x}{\ln(5)} \right]_0^1
\]

Substitute the limits:

\[
= \frac{5^1}{\ln(5)} - \frac{5^0}{\ln(5)}
\]

Since \( 5^1 = 5 \) and \( 5^0 = 1 \), this simplifies to:

\[
= \frac{5}{\ln(5)} - \frac{1}{\ln(5)}
\]

\[
= \frac{5 - 1}{\ln(5)} = \frac{4}{\ln(5)}
\]

### Final Answer:

\[
\int_0^1 5^x \, dx = \frac{4}{\ln(5)}
\]

Let me know if you need more details or further clarification!

### Related Questions:
1. How do you integrate \( a^x \) when the base is a variable?
2. What is the geometric interpretation of an exponential integral?
3. How can we approximate the value of \( \frac{4}{\ln(5)} \) numerically?
4. How does the integral of \( e^x \) compare with \( 5^x \)?
5. What is the derivative of the function \( 5^x \)?

**Tip**: When integrating exponential functions with arbitrary bases, always remember to divide by the natural logarithm of the base!

The integral \(\int_0^1 5^x \, dx\) equals?

Solve the integral \( \int_0^1 5^x \, dx \) with step-by-step instructions. Learn how to apply the exponential function integration formula and evaluate definite integrals, suitable for Grade 12 or college-level calculus students.

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