### Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation - AP Calc Study Guide

#### Introduction

Greetings, fellow calculus adventurers! Ready to embark on a journey to the mystical land of antiderivatives and indefinite integrals? 🎢 Think of this as the magical reverse spell to differentiation, where we turn derivatives back into smooth, continuous functions. Let’s dive in and unravel these enchanting mathematical concepts.

#### Indefinite Integrals: Notation

Before we jump into the wizardry of reversing derivatives, let’s talk about the notation and explore the mystical "family of functions." Imagine you have two different antiderivatives, like siblings F(x)=x²+3 and G(x)=x²−2; they both have the same derivative, 2x. If we reverse the derivative process through integration, we get 2x+C, where C is the magical constant 🎩✨, a.k.a. any constant ever! This gives rise to a whole family of functions differing only by C, but united by the same derivative. These are indefinite integrals because we can’t pinpoint which sibling (antiderivative) we’re dealing with unless we specify the bounds, as we would in definite integrals.

When writing this magical process, we use the notation: [ \int f(x)dx = F(x) + C ] Where F'(x) = f(x) and C denotes the integration constant.

#### Indefinite Integrals: Basic Rules

The adventure begins by reversing some of the derivatives we already know. Let’s go through the basic rules.

**Reverse Power Rule**
Just like a magician pulling a rabbit out of a hat, we're pulling exponents into integrals! For any function ( f(x) = \frac{x^{n+1}}{n+1} + C ) where ( n \neq -1 ) (because, let's face it, dividing by zero isn’t magical, it’s hazardous).

Here’s how it looks: [ \int x^n dx = \frac{x^{n+1}}{n+1} + C ]

**Example 1: Reverse Power Rule**

Evaluate (\int x^3 dx).

Using the reverse power rule: [ \int x^3 dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C ]

**Example 2: Reverse Power Rule with Fractions/Radicals**

Try this spicy one: [ \int \left(\frac{1}{x^2} - 7x^3 + 2x^2 - x + 4\right) dx ]

Rewrite first term: [ \int \left(x^{-2} - 7x^3 + 2x^2 - x + 4\right) dx ]

Now reverse power rule term by term: [ \int x^{-2} dx = -\frac{1}{x} ] [ \int -7x^3 dx = -\frac{7x^4}{4} ] [ \int 2x^2 dx = \frac{2x^3}{3} ] [ \int -x dx = -\frac{x^2}{2} ] [ \int 4 dx = 4x ]

Combine all: [ \int \left(\frac{1}{x^2} - 7x^3 + 2x^2 - x + 4\right) dx = -\frac{1}{x} - \frac{7x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 4x + C ]

#### Sums and Multiples Rules for Antiderivatives

Just like you can mix and match your favorite snacks, you can mix and match functions when integrating. 🍬+🍿=😁

**Sums Rule**: (\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx)**Multiples Rule**: (\int c \cdot f(x) dx = c \int f(x) dx)

**Example: Sums Rule**

[ \int [x^4 + x^2] dx = \int x^4 dx + \int x^2 dx ]

**Example: Multiples Rule**

[ \int 5x^6 dx = 5 \int x^6 dx ]

#### Antiderivatives of Trigonometric Functions

Time to switch our robes and dive into the realm of trigonometry. Ever wondered what functions sin(x) and cos(x) like to come from? Well, let’s find out! 😀

- (\int \sin(x) dx = -\cos(x) + C)
- (\int \cos(x) dx = \sin(x) + C)
- (\int \sec^2(x) dx = \tan(x) + C)
- (\int \csc^2(x) dx = -\cot(x) + C)
- (\int \sec(x) \tan(x) dx = \sec(x) + C)
- (\int \csc(x) \cot(x) dx = -\csc(x) + C)

#### Antiderivatives of Inverse Trig Functions

Not as common, but here are some to keep in your spellbook:

- (\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x) + C)
- (\int \frac{1}{1+x^2} dx = \tan^{-1}(x) + C)

#### Antiderivatives of Transcendental Functions

Finally, let's transcend the ordinary with these familiar faces:

- (\int \frac{1}{x} dx = \ln|x| + C)
- (\int e^x dx = e^x + C)

### Indefinite Integrals Practice Problems

Let's put your new skills to the test! 🧪

- Evaluate (\int x^7 dx)

Using the reverse power rule: [ \int x^7 dx = \frac{x^8}{8} + C = \boxed{\frac{1}{8}x^8 + C} ]

- Evaluate (\int [x^4 + \cos(x)] dx)

Split the integral: [ \int x^4 dx + \int \cos(x) dx ] [ = \frac{x^5}{5} + \sin(x) + C = \boxed{\frac{1}{5}x^5 + \sin(x) + C} ]

- Evaluate (\int [4\cos(x) + e^x] dx)

Split and simplify: [ 4\int \cos(x) dx + \int e^x dx ] [ = 4\sin(x) + e^x + C = \boxed{4\sin(x) + e^x + C} ]

- Evaluate (\int \left(\frac{3}{x} + x^2\right) dx)

Split and use appropriate rules: [ \int \frac{3}{x} dx + \int x^2 dx ] [ = 3 \ln|x| + \frac{x^3}{3} + C = \boxed{3 \ln|x| + \frac{x^3}{3} + C} ]

### Closing

Woah, what a ride! We’ve journeyed through reverse power rules, sums and multiples rules, trigonometric and transcendental functions. The biggest nugget of wisdom? When integrating, always add that magical constant "+C"! Now, go forth and integrate like a math wizard. 🧙♂️🧙♀️🍀

Good luck on your AP Calculus quest! You've got all the spells you need. 🚀