Topic 6.8 Notes – Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

1. What an Indefinite Integral Is

When you see

$$\int f(x)\,dx$$

you’re being asked:

“What function has derivative $f(x)$?”

If $F'(x) = f(x)$, then

$$\int f(x)\,dx = F(x) + C$$

• $F(x)$ is an antiderivative of $f(x)$.
• $C$ is an arbitrary constant.
• The result is a family of functions.

Why the +C matters

If $F'(x) = f(x)$, then
($F(x) + 7$)' = f(x) and ($F(x) - 3$)' = f(x) too.

All those functions differ by a constant, but have the same derivative.

On a quiz or FRQ, forgetting +C on an indefinite integral costs an easy point. Add it once at the very end, not after every term.

Big picture idea

• Differentiation asks for slope.
• Indefinite integration asks for the original function (up to a constant).

2. Core Rules for Finding Antiderivatives

Every rule here comes from reversing a derivative rule.

Reverse Power Rule

If $n \neq -1$,

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

What you do every time:

• Add 1 to the exponent.
• Divide by the new exponent.
• Add +C.

Example:

$$\int x^5 dx = \frac{x^6}{6} + C$$

The special case $n = -1$

The power rule would divide by 0, so it doesn’t work.

$$\int \frac{1}{x} dx = \ln |x| + C$$

That absolute value matters because $\ln(x)$ is only defined for $x>0$, but $\frac{1}{x}$ exists for negative $x$ too.

Rewrite Before You Integrate

This saves people constantly:

• $\frac{1}{x^3} = x^{-3}$
• $\sqrt{x} = x^{1/2}$
• $\frac{4}{\sqrt{x}} = 4x^{-1/2}$

Then use the power rule normally.

Sum Rule

$$\int [f(x) + g(x)] dx = \int f(x)dx + \int g(x)dx$$

You can integrate term by term.

Example:

$$\int (x^3 + 2x)dx = \frac{x^4}{4} + x^2 + C$$

Constant Multiple Rule

$$\int c \cdot f(x)dx = c \int f(x)dx$$

Pull constants out first.

Example:

$$\int 7x^2 dx = 7 \cdot \frac{x^3}{3} + C$$

3. Antiderivatives You Must Know Cold

Some functions don’t use the power rule. You just recognize them.

Trig Functions

Know these exactly:

• $\int \sin x \, dx = -\cos x + C$
• $\int \cos x \, dx = \sin x + C$
• $\int \sec^2 x \, dx = \tan x + C$
• $\int \csc^2 x \, dx = -\cot x + C$
• $\int \sec x \tan x \, dx = \sec x + C$
• $\int \csc x \cot x \, dx = -\csc x + C$

A common mistake is missing the negative sign on $\int \sin x$.

Exponential Function

$$\int e^x dx = e^x + C$$

It’s its own derivative, so it’s its own antiderivative.

Logarithmic Form

$$\int \frac{1}{x} dx = \ln |x| + C$$

Any time you see exactly $\frac{1}{x}$, think natural log.

Inverse Trig Patterns

These show up occasionally and are easy points if you recognize them:

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

You’re matching the pattern, not deriving it.

4. How to Approach an Indefinite Integral

When you see one:

1. Rewrite powers and radicals.
2. Separate sums.
3. Pull constants out.
4. Identify the rule.
5. Integrate.
6. Add +C once at the end.

If it looks like a basic derivative rule, reverse it.

Also know this: many functions do not have closed-form antiderivatives. On AP Calculus AB, if they expect you to find one, it will come from these standard rules.

5. Common Mistakes That Cost Points

• Using the power rule on $\frac{1}{x}$.
• Dividing by the old exponent instead of the new one.
• Forgetting the negative sign for trig.
• Writing $\ln x$ instead of $\ln |x|$.
• Forgetting +C.

Multiple choice often tests small algebra slips, not big ideas.