\int \frac{2x}{x^2+5x}

Show me the full steps how to solve it

To solve the integral \(\int \frac{2x}{x^2+5x} \, dx\), we can follow these steps:

### Step 1: Simplify the Integrand
First, let's factor the quadratic expression in the denominator:

\[
x^2 + 5x = x(x + 5)
\]

So the integral becomes:

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

### Step 2: Simplify the Fraction
The fraction \(\frac{2x}{x(x + 5)}\) can be simplified by canceling out \(x\) in the numerator and denominator:

\[
\frac{2x}{x(x + 5)} = \frac{2}{x + 5}
\]

Now the integral simplifies to:

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

### Step 3: Integrate Using a Simple Substitution
This integral is now in a simple form that can be directly integrated. The integral of \(\frac{1}{x + a}\) is \(\ln|x + a|\). So:

\[
\int \frac{2}{x + 5} \, dx = 2 \int \frac{1}{x + 5} \, dx = 2 \ln|x + 5| + C
\]

### Step 4: Write the Final Answer
Thus, the final answer to the integral is:

\[
\boxed{2 \ln|x + 5| + C}
\]

Where \(C\) is the constant of integration.

---

Would you like any additional details or have any questions about these steps? Here are some related questions you might explore:

1. How do you integrate \(\int \frac{1}{x^2+5x} \, dx\)?
2. How do you solve \(\int \frac{2x}{(x+1)(x+5)} \, dx\)?
3. Can you integrate \(\int \frac{1}{x(x+5)} \, dx\) using partial fractions?
4. What is the integral of \(\int \frac{1}{x^2+5} \, dx\)?
5. How do you integrate \(\int \frac{x}{x^2+5x+6} \, dx\)?
6. What is the integral of \(\int \frac{x^2}{x^2+5x} \, dx\)?
7. Can you explain the substitution method for \(\int \frac{1}{x + 5} \, dx\)?
8. What happens if you substitute \(u = x + 5\) before integrating?

**Tip:** Always check if the integrand can be simplified before attempting to integrate—it often makes the process much easier!

Learn how to solve the integral \(\int \frac{2x}{x^2+5x} \, dx\) with detailed step-by-step instructions. This problem involves calculus concepts, integration techniques, and rational functions. Suitable for college-level students and anyone interested in advanced calculus topics.

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