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The problem you provided is:

\[
I = \int \frac{3x + 2}{x^2 + 5} \, dx
\]

Let's solve this integral step by step.

### Step 1: Break the integral into two parts.
We can rewrite the integrand as:

\[
\frac{3x + 2}{x^2 + 5} = \frac{3x}{x^2 + 5} + \frac{2}{x^2 + 5}
\]

Thus, the integral becomes:

\[
I = \int \frac{3x}{x^2 + 5} \, dx + \int \frac{2}{x^2 + 5} \, dx
\]

### Step 2: Solve the first integral.

For the first integral:

\[
\int \frac{3x}{x^2 + 5} \, dx
\]

Use substitution: let \( u = x^2 + 5 \), so \( du = 2x \, dx \). Adjusting for the 3:

\[
\int \frac{3x}{x^2 + 5} \, dx = \frac{3}{2} \int \frac{du}{u} = \frac{3}{2} \ln|u| = \frac{3}{2} \ln(x^2 + 5)
\]

### Step 3: Solve the second integral.

For the second integral:

\[
\int \frac{2}{x^2 + 5} \, dx
\]

This is a standard integral, which is solved using the formula:

\[
\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right)
\]

In this case, \( a^2 = 5 \), so \( a = \sqrt{5} \). Thus, the integral becomes:

\[
\int \frac{2}{x^2 + 5} \, dx = \frac{2}{\sqrt{5}} \tan^{-1}\left(\frac{x}{\sqrt{5}}\right)
\]

### Step 4: Combine the results.

Now, combining both parts:

\[
I = \frac{3}{2} \ln(x^2 + 5) + \frac{2}{\sqrt{5}} \tan^{-1}\left(\frac{x}{\sqrt{5

Explore a detailed step-by-step solution for the integral ∫ (3x + 2) / (x^2 + 5) dx. This problem involves breaking down the integral into manageable parts and applying key calculus formulas. Suitable for advanced high school or college students.

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