Evaluate the definite integral.
28
x

25 + 2x
 dx

0


The definite integral you have provided is:

\[
\int_0^{28} \frac{x}{25 + 2x} \, dx
\]

To solve this, we'll use substitution.

### Step 1: Substitution
Let:
\[
u = 25 + 2x
\]
Then:
\[
du = 2 \, dx \quad \text{or} \quad \frac{du}{2} = dx
\]

When \( x = 0 \), \( u = 25 \), and when \( x = 28 \), \( u = 25 + 2(28) = 81 \).

### Step 2: Rewrite the integral
Substituting into the original integral:

\[
\int_0^{28} \frac{x}{25 + 2x} \, dx = \int_{25}^{81} \frac{\frac{u - 25}{2}}{u} \cdot \frac{du}{2}
\]

Simplifying this expression:

\[
= \frac{1}{4} \int_{25}^{81} \left( 1 - \frac{25}{u} \right) du
\]

### Step 3: Break into two integrals
Now we can split the integral:

\[
= \frac{1}{4} \left( \int_{25}^{81} du - 25 \int_{25}^{81} \frac{1}{u} du \right)
\]

### Step 4: Solve each integral
The first integral is straightforward:

\[
\int_{25}^{81} du = 81 - 25 = 56
\]

The second integral is a logarithmic integral:

\[
\int_{25}^{81} \frac{1}{u} du = \ln|u|\Big|_{25}^{81} = \ln(81) - \ln(25)
\]

### Step 5: Combine results
Now, putting everything together:

\[
\frac{1}{4} \left( 56 - 25 \left( \ln(81) - \ln(25) \right) \right)
\]

Simplifying further:

\[
= \frac{1}{4} \left( 56 - 25 \ln\left(\frac{81}{25}\right) \right)
\]

This is the exact value of the integral. For a numerical approximation:

\[
\ln\left(\frac{81}{25}\right) = \ln(3.24) \approx 1.176
\]

Thus:

\[
\approx \frac{1}{4} \left( 56 - 25 \times 1.176 \right)
= \frac{1}{4} \left( 56 - 29.4 \right)
= \frac{1}{4} \times 26.6
= 6.65
\]

### Final Answer:
The value of the definite integral is approximately:

\[
6.65
\]

Would you like more details or have any questions?

### Related Questions:
1. How can substitution simplify integration?
2. What are other methods to evaluate definite integrals?
3. Why does the logarithmic function arise in integrals of rational expressions?
4. How do definite integrals differ from indefinite integrals?
5. Can integration be used to find areas under curves?

### Tip:
When dealing with rational expressions, substitution often simplifies the integration by transforming the integral into a simpler form.

Evaluate the definite integral: ∫(x / (25 + 2x)) dx from 0 to 28.

Learn how to evaluate the definite integral of x / (25 + 2x) from 0 to 28 using substitution. This step-by-step solution covers key concepts in integration, substitution, and logarithmic functions, resulting in a final answer of approximately 6.65.

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