6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

An antiderivative reverses differentiation: if $F'(x)=f(x)$, then $F(x)$ is an antiderivative of $f$. The indefinite integral $\int f(x)\,dx=F(x)+C$ captures the whole family of antiderivatives, so you always add the constant of integration $C$. For AP Calculus, include $+C$ for indefinite integrals and leave it off when evaluating definite integrals.

Why This Matters for the AP Calculus Exam

Antiderivatives are the engine behind almost every integral you will compute in AP Calculus. The Fundamental Theorem of Calculus tells you that to evaluate a definite integral, you first need an antiderivative, so the basic rules here feed directly into definite integral problems, accumulation functions, differential equations, and area and volume work later in the course.

On both multiple-choice and free-response questions, you will need to recognize patterns quickly and write antiderivatives cleanly. When you evaluate an indefinite integral, including the $+C$ is part of correct work. Clear notation and accurate antiderivatives are important for showing reasoning that graders and your future self can follow.

Key Takeaways

• $\int f(x)\,dx = F(x) + C$ means $F'(x) = f(x)$, and $C$ can be any constant.
• Every basic antiderivative comes from reversing a derivative rule you already know.
• Use the reverse power rule $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
• Integration is linear: you can split sums and pull out constant multiples.
• Memorize the standard trig, inverse-trig, and exponential antiderivatives.
• Some functions, like $e^{x^2}$, have no closed-form antiderivative, so do not force an algebraic answer.

Indefinite Integrals: Notation

Start with a family of functions before reversing the derivative process.

Imagine two different antiderivatives, $F(x) = x^2+3$ and $G(x) = x^2-2$.

Take the derivative of both, and they share the same derivative, $2x$. When you reverse the process through integration, how do you account for arriving at these two different antiderivatives? That is the job of the constant $C$.

When you integrate $2x$, the antiderivative is $2x+C$ where $C$ is any constant. This result is called a family of functions because they differ only in their constant and all share the same derivative.

This is an indefinite integral because you can not tell which member of the family is intended. If no bounds are given (unlike a definite integral), always add $+C$ to the end of your antiderivative.

Here is the general notation:

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

Where $F'(x)=f(x)$ and $C$ represents the constant of integration.

Indefinite Integrals: Basic Rules

Now look at how to reverse some of the derivatives from earlier in the course.

Reverse Power Rule

The reverse power rule undoes the power rule from differentiation. Suppose you have:

$$f(x) = \frac{x^{n+1}}{n+1}+C$$

Where $n \neq -1$, since $n=-1$ makes $f(x)$ undefined.

What is its derivative? Using the power rule for derivatives,

$$f'(x) = x^{n}$$

Now reverse it. Since antiderivatives and derivatives are inverses,

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

This is the reverse power rule. You add one to the exponent and divide by the new exponent.

Reverse Power Rule Example 1

Evaluate:

$$\int x^3dx$$

Using the reverse power rule,

$$\int x^3dx=\frac{x^{3+1}}{3+1}+C=\frac{x^{4}}{4}+C$$

Reverse Power Rule Example 2

Try this one. A useful tip is to rewrite fractions with negative exponents. The same idea works for radicals, since they can be rewritten with fractional exponents.

$$\int(\frac{1}{x^2}-7x^3+2x^2-x+4)\, dx$$

Rewrite the first term:

$$\int(x^{-2}-7x^3+2x^2-x+4)\, dx$$

Apply the reverse power rule term by term:

$$\int(\frac{1}{x^2}-7x^3+2x^2-x+4)\, dx = -\frac{1}{x} -\frac{7x^4}{4}+\frac{2x^3}{3}-\frac{x^2}{2}+4x+C$$

Sums and Multiples Rules for Antiderivatives

Just as derivatives had sum and constant-multiple rules in Unit 2, antiderivatives have matching rules.

The sums rule states that

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

The multiples rule states that

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

Sums and Multiples Examples

The first shows the sums rule and the second shows the multiples rule.

$$\int \left[x^4+x^2\right]dx=\int x^4dx + \int x^2dx$$ $$\int 5x^{6}dx=5\int x^6dx$$

Antiderivatives of Trigonometric Functions

When you are first learning trig antiderivatives, ask yourself, "What has a derivative of this?"

Antiderivative of $\sin(x)$

Recall that $\frac{d}{dx}[\cos(x)]=-\sin(x)$. This means $\frac{d}{dx}[-\cos(x)]=\sin(x)$. Therefore,

$$\int \sin(x)dx=-\cos(x)+C$$

Antiderivative of $\cos(x)$

Recall that $\frac{d}{dx}[\sin(x)]=\cos(x)$. Therefore,

$$\int \cos(x)dx=\sin(x)+C$$

Other Antiderivatives of Trig Functions

Know these trig integrals for the AP Calculus exam:

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

Antiderivatives of Inverse Trig Functions

These appear less often, but here are the forms you may see:

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

Antiderivatives of Transcendental Functions ($\frac{1}{x}$, $e^x$)

Finally, here are integrals for the transcendental functions you are likely to encounter.

Antiderivative of $\frac{1}{x}$

Recall that $\frac{d}{dx}[\ln(x)]=\frac{1}{x}$. So a first guess is $\int \frac{1}{x}dx=\ln(x)+C$.

However, because the domain of $\ln(x)$ is $(0, \infty)$, if you want an antiderivative of $\frac{1}{x}$ for any positive or negative $x$, rewrite the rule as

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

Antiderivative of $e^{x}$

Recall that $\frac{d}{dx}[e^{x}]=e^{x}$. Therefore,

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

How to Use This on the AP Calculus Exam

Problem Solving

• Read the integrand and ask which derivative rule produces it. That recognition is the whole strategy for basic antiderivatives.
• Rewrite before you integrate. Turn $\frac{1}{x^2}$ into $x^{-2}$ and $\sqrt{x}$ into $x^{1/2}$ so the reverse power rule applies cleanly.
• Split sums and pull out constants using the linearity rules, then integrate each piece.

Checking Your Work

• Differentiate your answer. If you get back the original integrand, your antiderivative is correct.
• Watch the sign on $\int \sin(x)\,dx = -\cos(x)+C$. This is the most common sign slip.

Common Trap

• Always write $+C$ on an indefinite integral. Leaving it off changes a family of functions into a single function and is incomplete work.

Indefinite Integrals Practice Problems

Now that you know the basic rules, try some practice problems.

Indefinite Integrals Problems

Evaluate each integral.

$$1. \int x^7dx=?$$ $$2.\int \left[x^4+\cos(x)\right]dx=?$$ $$3.\int \left[4\cos(x)+e^x\right]dx=?$$ $$4.\int (\frac{3}{x}+x^2) \, dx$$

Indefinite Integrals Question Solutions

Indefinite Integrals Question 1

This one calls for the reverse power rule.

$$\int x^7dx=\frac{x^{7+1}}{7+1}+C=\boxed{\frac{1}{8}x^{8}+C}$$

Indefinite Integrals Question 2

Using the sums rule,

$$\int \left[x^4+\cos(x)\right]dx=\int x^4dx+\int \cos(x)dx$$

Take the antiderivatives separately, then add them.

Using the reverse power rule,

$$\int x^4dx = \frac{x^{4+1}}{4+1}+C=\frac{1}{5}x^5 + C$$

And using the antiderivative of $\cos(x)$,

$$\int \cos(x)dx = \sin(x)+C$$

Combining,

$$\int \left[x^4+\cos(x)\right]dx=\boxed{\frac{1}{5}x^5 +\sin(x)+ C}$$

Indefinite Integrals Question 3

Using the sums rule,

$$\int \left[4\cos(x)+e^x\right]dx=\int 4\cos(x)dx + \int e^xdx$$

Using the multiples rule and the antiderivative of $\cos(x)$,

$$\int 4\cos(x)dx=4\int \cos(x)dx=4\sin(x)+C$$

And using the antiderivative of $e^x$,

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

Combining,

$$\int \left[4\cos(x)+e^x\right]dx=\boxed{4\sin(x)+e^x+C}$$

Indefinite Integrals Question 4

Using the sums rule,

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

Take the integral of the first term using $\int \frac{1}{x}dx=\ln\mid x \mid + C$:

$$∫ (\frac{3}{x}) dx = 3 ln(|x|) + C$$

And using the reverse power rule,

$$∫ x^2 dx = \frac{x^3}{3} + C$$

Combining,

$$\int (\frac{3}{x}+x^2) \, dx =\boxed{3ln(|x|) + \frac{x^3}{3}+C}$$

Common Misconceptions

• Forgetting $+C$. An indefinite integral represents a whole family of functions, so the constant of integration is required. Definite integrals do not carry a $+C$ because the constant cancels.
• Misusing the reverse power rule on $\frac{1}{x}$. You can not apply $\frac{x^{n+1}}{n+1}$ when $n=-1$, since that divides by zero. Instead, $\int \frac{1}{x}\,dx = \ln|x| + C$, with absolute value bars so it works for negative $x$ too.
• Dropping the negative sign on $\int \sin(x)\,dx$. The antiderivative of $\sin(x)$ is $-\cos(x)+C$, not $\cos(x)+C$. Differentiate to confirm.
• Assuming every function has a closed-form antiderivative. Some integrals, such as $\int e^{x^2}\,dx$ and $\int \sin(x^2)\,dx$, can not be written with elementary functions. Recognizing this saves you from chasing an answer that does not exist in basic form.
• Treating integration as just reversing steps mechanically. You have to recognize which rule fits the integrand. Rewriting the function first often reveals the right pattern.

Related AP Calculus Guides

• Unit 6 Overview: Integration and Accumulation of Change
• 6.11 Integrating Using Integration by Parts
• 6.1 Integration and Accumulation of Change
• 6.12 Integrating Using Linear Partial Fractions
• 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
• 6.4 The Fundamental Theorem of Calculus and Accumulation Functions