# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/integral/properties-of-indefinite-integral
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/integral/properties-of-indefinite-integral/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Properties of Indefinite Integral",
    description: "Master indefinite integral rules: power rule, constant multiples, sum/difference rules, and u-substitution techniques with step-by-step examples.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "05/26/2025",
    subject: "Integrals",
};

## Power Rule

This is one of the most basic and important rules in integration. If you encounter an integral in power form, use this formula:

<BlockMath math="\int x^n \, dx = \frac{1}{n+1}x^{n+1} + C, \quad \text{where } n \neq -1" />

Simply put, add one to the exponent, then divide by that new exponent.

For example, let's solve <InlineMath math="\int x^5 \, dx" />.

Here, <InlineMath math="n=5" />, so:

<div className="flex flex-col gap-4">
<BlockMath math="\int x^5 \, dx = \frac{1}{5+1}x^{5+1} + C" />
<BlockMath math="= \frac{1}{6}x^6 + C" />
</div>

## Constant Multiple Rule

If the function to be integrated has a coefficient or constant, you can "pull" that constant out of the integral to simplify the calculation.

<BlockMath math="\int a \cdot f(x) \, dx = a \int f(x) \, dx" />

> Remember, **a** is a constant number. This property is very useful for simplifying an integral before solving it.

## Sum and Difference Rule

What if we have to integrate two functions that are added or subtracted? Easy, we can separate them into two different integrals.

<BlockMath math="\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx" />

For example, to solve <InlineMath math="\int (x^4 - x^3) \, dx" />, we first separate them:

<div className="flex flex-col gap-4">
<BlockMath math="\int (x^4 - x^3) \, dx = \int x^4 \, dx - \int x^3 \, dx" />
<BlockMath math="= \left( \frac{1}{4+1}x^5 \right) - \left( \frac{1}{3+1}x^4 \right) + C" />
<BlockMath math="= \frac{1}{5}x^5 - \frac{1}{4}x^4 + C" />
</div>

Don't forget to add only one constant **C** at the end. Technically, each integral would produce its own constant (e.g., <InlineMath math="C_1" /> and <InlineMath math="C_2" />). However, since all these constants are of an indeterminate value, their sum or difference will also result in a new indeterminate constant. Therefore, we just write it as a single **C** at the end.

## Substitution Rule

For integrals that look complicated, like the product of two functions where one is the derivative of the other (or a multiple of it), we can use the **substitution rule**. The basic idea is to simplify the integral by replacing part of the function with a new variable, usually **u**.

<BlockMath math="\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du" />

This is often called **u-substitution**. This method is very powerful for solving complex integrals.

## Exercises

1.  Find the result of <InlineMath math="\int 6x^2 \, dx" />!
2.  Find the result of <InlineMath math="\int (3x^2 + 2x - 5) \, dx" />!
3.  Solve the integral <InlineMath math="\int 2x(x^2 + 3)^4 \, dx" /> using the substitution rule!

### Answer Key

1.  To solve <InlineMath math="\int 6x^2 \, dx" />, we can use the **Constant Multiple Rule** and the **Power Rule**.
    
    **Step 1:** Pull the constant 6 out of the integral.
    
    <BlockMath math="\int 6x^2 \, dx = 6 \int x^2 \, dx" />
    
    **Step 2:** Use the power rule on <InlineMath math="\int x^2 \, dx" />, where <InlineMath math="n=2" />.
    
    <div className="flex flex-col gap-4">
    <BlockMath math="6 \left( \frac{1}{2+1}x^{2+1} \right) + C" />
    <BlockMath math="= 6 \left( \frac{1}{3}x^3 \right) + C" />
    </div>
    
    **Step 3:** Multiply the constants to get the final result.
    
    <BlockMath math="= 2x^3 + C" />

2.  For the integral <InlineMath math="\int (3x^2 + 2x - 5) \, dx" />, we use the **Sum and Difference Rule** to break it down into three separate integrals.
    
    **Step 1:** Separate each term into its own integral.
    
    <BlockMath math="\int 3x^2 \, dx + \int 2x \, dx - \int 5 \, dx" />
    
    **Step 2:** Solve each integral one by one using the power and constant rules.
    
    <div className="flex flex-col gap-4">
    <BlockMath math="= 3\left(\frac{1}{3}x^3\right) + 2\left(\frac{1}{2}x^2\right) - 5x + C" />
    <BlockMath math="= x^3 + x^2 - 5x + C" />
    </div>
    
    So, the answer is <InlineMath math="x^3 + x^2 - 5x + C" />.

3.  The integral <InlineMath math="\int 2x(x^2 + 3)^4 \, dx" /> is a classic example for the **Substitution Rule**.
    
    **Step 1:** Choose a part of the function to be <InlineMath math="u" />. A good choice is the part inside the parentheses.
    
    Let: <InlineMath math="u = x^2 + 3" />.
    
    **Step 2:** Find the derivative of <InlineMath math="u" /> with respect to <InlineMath math="x" />, which is <InlineMath math="du/dx" />.
    
    <BlockMath math="\frac{du}{dx} = 2x" />
    
    From this, we can write <InlineMath math="du = 2x \, dx" />.
    
    **Step 3:** Perform the substitution. Replace <InlineMath math="x^2+3" /> with <InlineMath math="u" /> and <InlineMath math="2x \, dx" /> with <InlineMath math="du" />.
    
    <BlockMath math="\int (x^2 + 3)^4 (2x \, dx) = \int u^4 \, du" />
    
    **Step 4:** Solve the simplified integral using the power rule.
    
    <BlockMath math="\int u^4 \, du = \frac{1}{5}u^5 + C" />
    
    **Step 5:** Substitute <InlineMath math="u" /> back to its original form.
    
    <BlockMath math="= \frac{1}{5}(x^2 + 3)^5 + C" />
    
    This is the final result.