https://file.aiquickdraw.com/mathbot/file/1730118305931hskuo10w.jpg

The image contains a mathematical expression, which appears to be:

\[
\int \frac{2x + 1}{x^2 - 5} \, dx
\]

To solve this integral, we will use substitution or partial fraction decomposition. Here's a step-by-step solution:

### Step 1: Check if substitution is appropriate
Let’s try a substitution where \( u = x^2 - 5 \), making \( du = 2x \, dx \).

This substitution simplifies the numerator \( 2x \), but we still have the \( +1 \) term that needs handling. So, substitution alone won't simplify the expression adequately.

### Step 2: Split the integral
Instead, split the original expression as:
\[
\int \frac{2x + 1}{x^2 - 5} \, dx = \int \frac{2x}{x^2 - 5} \, dx + \int \frac{1}{x^2 - 5} \, dx
\]

Now we have two separate integrals to evaluate.

### Step 3: Solve the first integral
For the integral \( \int \frac{2x}{x^2 - 5} \, dx \), use the substitution \( u = x^2 - 5 \), giving \( du = 2x \, dx \), resulting in:
\[
\int \frac{2x}{x^2 - 5} \, dx = \int \frac{1}{u} \, du = \ln |u| + C = \ln |x^2 - 5| + C
\]

### Step 4: Solve the second integral
The second integral is \( \int \frac{1}{x^2 - 5} \, dx \). This can be approached using a trigonometric substitution:
Let \( x = \sqrt{5} \tan \theta \), then \( dx = \sqrt{5} \sec^2 \theta \, d\theta \), and \( x^2 - 5 = 5 (\tan^2 \theta - 1) = 5 \sec^2 \theta \), transforming the integral into a form solvable by trigonometric identities.

Would you like more details on solving the second integral or have any other questions? 

Here are five related questions to expand on this topic:
1. How do we solve integrals involving quadratic denominators without real roots?
2. What are the conditions for using partial fraction decomposition in integrals?
3. Can we solve this integral using numerical methods? How would the result compare?
4. How do hyperbolic functions relate to integrals with quadratic terms?
5. How does changing variables simplify different types of integrals?

**Tip:** Always check if a given integral can be simplified by substitution or splitting it into more manageable parts before using more advanced techniques.

This tutorial provides a detailed solution for the integral ∫(2x + 1) / (x^2 - 5) dx. It demonstrates the use of substitution, partial fractions, and trigonometric substitution techniques, suitable for advanced high school and college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation