Teach me step by step how to integrate an algebraic expression

Integrating an algebraic expression involves finding the antiderivative, or the integral, of a given function with respect to a variable. Below is a step-by-step guide on how to integrate a basic algebraic expression.

### Step 1: Identify the Basic Form
An algebraic expression typically consists of terms of the form \( ax^n \), where:
- \( a \) is a constant coefficient.
- \( x \) is the variable.
- \( n \) is the exponent of \( x \).

### Step 2: Apply the Power Rule of Integration
The power rule for integration states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \( n \neq -1 \))}
\]
Here, \( C \) is the constant of integration, which accounts for any constant that might have been differentiated away.

### Step 3: Integrate Each Term Separately
For an expression like \( ax^n \), apply the power rule and multiply by the constant \( a \):
\[
\int ax^n \, dx = a \cdot \frac{x^{n+1}}{n+1} + C
\]

### Example 1: Integrating a Simple Expression
Let’s integrate the expression \( 3x^2 \):
1. Identify the term: \( 3x^2 \), where \( a = 3 \) and \( n = 2 \).
2. Apply the power rule:
   \[
   \int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C
   \]
3. Simplify:
   \[
   \int 3x^2 \, dx = x^3 + C
   \]

### Example 2: Integrating a Polynomial
Consider integrating \( 2x^3 - 5x + 4 \):
1. Break the expression into separate terms:
   \[
   \int (2x^3 - 5x + 4) \, dx = \int 2x^3 \, dx - \int 5x \, dx + \int 4 \, dx
   \]
2. Integrate each term using the power rule:
   \[
   \int 2x^3 \, dx = 2 \cdot \frac{x^{3+1}}{3+1} = \frac{2x^4}{4} = \frac{x^4}{2}
   \]
   \[
   \int -5x \, dx = -5 \cdot \frac{x^{1+1}}{1+1} = -5 \cdot \frac{x^2}{2} = -\frac{5x^2}{2}
   \]
   \[
   \int 4 \, dx = 4x
   \]
3. Combine the results:
   \[
   \int (2x^3 - 5x + 4) \, dx = \frac{x^4}{2} - \frac{5x^2}{2} + 4x + C
   \]

### Step 4: Combine the Results
If you have multiple terms, simply add up the integrals of each term to get the final integrated expression.

### Final Example for Practice
Integrate \( 7x^5 - 2x^3 + 6x^2 - 4x + 8 \):
1. Break down the expression and apply the power rule term by term.
2. Integrate:
   \[
   \int 7x^5 \, dx = \frac{7x^6}{6}
   \]
   \[
   \int -2x^3 \, dx = -\frac{2x^4}{4} = -\frac{x^4}{2}
   \]
   \[
   \int 6x^2 \, dx = 2x^3
   \]
   \[
   \int -4x \, dx = -2x^2
   \]
   \[
   \int 8 \, dx = 8x
   \]
3. Combine:
   \[
   \int (7x^5 - 2x^3 + 6x^2 - 4x + 8) \, dx = \frac{7x^6}{6} - \frac{x^4}{2} + 2x^3 - 2x^2 + 8x + C
   \]

Would you like more details on any step or have any questions?

### 5 Related Questions:
1. How do you integrate an expression with a negative exponent?
2. What is the process for integrating a trigonometric function?
3. How do you handle integration by parts?
4. What changes when integrating an expression with a square root?
5. How do you approach definite integrals?

### Tip:
Always remember to add the constant of integration \( C \) at the end of indefinite integrals.

Learn how to integrate algebraic expressions step-by-step with detailed examples. This guide covers the power rule of integration and provides practice examples for integrating polynomials. Suitable for advanced high school and college students.

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