Unit 6Antiderivatives for Basic Functions

Concept 4.109Built for May 2027 AP Exam

Antiderivatives for Basic Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

An antiderivative reverses a derivative, and the reverse power rule is the engine: raise the exponent by one and divide by the new exponent. The exception is $x^{-1}$ , whose antiderivative is the natural logarithm, because raising $-1$ by one gives zero and division by zero is unavailable. Sine, cosine, and the exponential each reverse with a sign to track or none at all. Every indefinite integral carries a constant, since shifting a function vertically leaves every slope unchanged. On the exam the skill rarely appears as its own part; it sits inside evaluation problems, where a wrong antiderivative destroys a computation that was otherwise correct.

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Mostly indirect use. A no-calculator part asks for a total from a polynomial rate, and the antiderivative is the first step. Occasionally an item asks directly for $\int f(x)\,dx$ and offers four candidate families.
What you’ll do: Reverse one rule at a time, check the result by differentiating it back, and keep the constant when the integral has no limits.
Memory hook: Up one, divide by the new one. The exponent rises by one and the same number lands underneath.
Use it on the exam: For MCQs: Differentiate the choices instead of integrating. Only one candidate will return the integrand, and the check takes less time than the computation. For FRQs: When the antiderivative is a step inside a larger part, differentiate your answer in the margin before you use it. One line of checking prevents an error that would otherwise cost every point in the part.

Concept Review

A Family, Not a Function

Figure 1 — Every curve in the family has the same slope at the same $x$ , so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

If $F'(x) = f(x)$ on an interval, then $F$ is an antiderivative of $f$ there, and so is $F(x) + C$ for any constant $C$ . The indefinite integral records the whole family:

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

A definite integral does not need the constant, because the two evaluations cancel it. An indefinite integral does, and dropping it is a scoring error rather than a stylistic one.

Every curve in the family has the same slope at the same $x$, so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

The Rules Worth Memorizing

Integrand	Antiderivative	Check by differentiating
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	$(n+1)\frac{x^{n}}{n+1} = x^{n}$
$x^{-1} = \frac{1}{x}$	$\ln\lvert x \rvert + C$	$\frac{1}{x}$
$k$ , a constant	$kx + C$	$k$
$e^{x}$	$e^{x} + C$	$e^{x}$
$\cos x$	$\sin x + C$	$\cos x$
$\sin x$	$-\cos x + C$	$\sin x$
$\sec^{2} x$	$\tan x + C$	$\sec^{2} x$

Two of these deserve a note. The absolute value in $\ln\lvert x \rvert$ keeps the antiderivative defined on intervals where $x$ is negative, and the exam accepts either form when the domain is positive. The minus sign in $-\cos x$ is the most frequently dropped symbol in the whole unit, and it is invisible in a final answer that is only checked for plausibility.

Linear Inner Functions

The chain rule leaves a trace that can be reversed when the inner function is linear:

$\int f(ax + b)\,dx = \frac{1}{a}F(ax + b) + C$

So $\int e^{3x}\,dx = \frac{1}{3}e^{3x} + C$ and $\int \cos(2x)\,dx = \frac{1}{2}\sin(2x) + C$ . The factor is the reciprocal of the inner slope, and it is easy to test: differentiate the answer and watch the factor cancel. When the inner function is not linear, this shortcut fails, which is why $\int e^{x^2}\,dx$ has no elementary form at all.

Functions With No Elementary Antiderivative

Many continuous functions cannot be antidifferentiated in closed form. The standard examples are $e^{-x^2}$ , $\sin\left(x^2\right)$ , $\frac{\sin x}{x}$ , and $\sqrt{1 + x^3}$ . This is not a gap in your technique; it is a theorem about which elementary expressions have elementary integrals. On the exam these appear only inside definite integrals, where a graph, a table, or a calculator supplies the value, and the correct response is never to search for a formula.

Checking Is a One-Line Habit

Every antiderivative can be verified by differentiating it. The habit costs a single line and catches sign errors, exponent errors, and missing reciprocal factors at once. In a multi-step FRQ the check also protects the later parts, because an antiderivative that is wrong by a constant multiple silently scales every subsequent answer.

Common Misconception Check

Myth 1: “ $\int x^{-1}\,dx = \frac{x^{0}}{0}$ .” Truth: That expression is undefined. The antiderivative of $x^{-1}$ is $\ln\lvert x \rvert$ , and this is the single exception the power rule cannot cover.
Myth 2: “The $+C$ is decoration.” Truth: It is the difference between a family and a member. Any part that asks for a particular solution depends on it.
Myth 3: “Every continuous function has an antiderivative you can write down.” Truth: Every continuous function has an antiderivative. Very few have one built from elementary functions.

Topic-Specific Study Method

Reverse One Rule, Then Check

Borrowed idea: Reverse the rules, then check. Antidifferentiation is differentiation read backwards. The table is the same, and the direction is the only change. Raise the exponent, divide by the new number, and differentiate to confirm. One line of checking beats ten of hoping. The check catches sign errors, exponent errors, and missing reciprocal factors at once, and it costs less time than rereading the work you have just written.

Exam disguise	What you see on the page	What your brain must translate	Required execution
“Evaluate the integral” with a polynomial	powers and coefficients	reverse power rule	raise, divide, check
A rate given as a polynomial	“how many words by the end”	antiderivative then subtract	constant term becomes $kx$
Choices that are functions	four candidate families	differentiate each	pick the one returning the integrand
$\frac{1}{x}$ in the integrand	a negative power	the logarithm case	absolute value, plus $C$
A composite like $e^{3x}$	a linear inner function	reciprocal factor	divide by the inner slope

① Raise the exponent before doing anything else. Writing the new exponent down first prevents the classic slip of dividing by the old one.

② Handle the constant term separately. A bare number integrates to that number times $x$ , not to zero and not to a higher power.

③ Differentiate the answer. If the result is not the integrand, the error is in the exponent or in a sign, and both are visible in one line.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\int\left(6x^{2} - \frac{4}{x} + 5\right)dx$ .

Standard Execution:

Power term. $\int 6x^{2}\,dx = 2x^{3}$ .

Logarithm term. $\int \frac{4}{x}\,dx = 4\ln\lvert x \rvert$ .

Constant term. $\int 5\,dx = 5x$ .

Assemble. $2x^{3} - 4\ln\lvert x \rvert + 5x + C$ .

Check. Differentiating gives $6x^{2} - \frac{4}{x} + 5$ , which matches.

Exam Variation (Context / Parameter Shift): A teacher reads at a rate modeled by $W(t) = -3t^{2} + 8t + 100$ words per minute for $0 \le t \le 10$ . Find the total number of words read in the ten minutes.

Solution: An antiderivative is $F(t) = -t^{3} + 4t^{2} + 100t$ . Then $F(10) = -1000 + 400 + 1000 = 400$ and $F(0) = 0$ , so the total is $400$ words. Two checks are worth doing: the units are words because a rate in words per minute was integrated over minutes, and differentiating $F$ returns $W$ exactly, which confirms that the constant term became $100t$ rather than disappearing.

Common Traps & Failure Signals:

Dividing by the old exponent (Signal: a coefficient of $6x^{3}$ from $6x^{2}$ ): divide by the new exponent, which is the one you just wrote.
Constant term vanished (Signal: an antiderivative with no linear term when the integrand had a constant): a constant integrates to $kx$ .
Missing constant (Signal: an indefinite integral with no $+C$ ): the family needs it.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Exponent Off By One: Write the new exponent first.
Sign Dropped: You integrated $\sin x$ to $\cos x$ and lost the minus, which is not a memory lapse but a direction error, since the derivative of $\cos$ carries the minus and reversing must carry it too.
Constant Ignored: You treated a bare number as if it integrated to nothing, so the antiderivative is off by a linear term that looks harmless on a graph and changes the value of every definite integral built on it.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Two of the nine released exams since 2017 contain a part in which producing an antiderivative from the basic rules is the scored step, and both are no-calculator parts
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; The skill otherwise appears inside other parts and dominates the multiple-choice section, where the reverse power rule is tested directly
FRQ Years Appeared: 2019 (Q3b), 2025 (Q3D) ---

Practice Examples

MCQ 1

The Reverse Power Rule

$\int\left(8x^{3} - 6x\right)dx =$

MCQ 2

The Logarithm Case

$\int \frac{5}{x}\,dx =$

MCQ 3

A Composite With a Linear Inside

$\int \cos(4x)\,dx =$

FRQ 4

From a Rate to a Total

A pump moves water at a rate modeled by $r(t) = 12 - 3t^{2}$ gallons per minute for $0 \le t \le 2$ minutes.

(a) Find $\int r(t)\,dt$ .

(b) Find the total volume of water moved during the two minutes.

(c) A student claims that $\int_0^{2} r(t)\,dt$ is the area of the region between the graph of $r$ and the $t$ -axis. Explain why the claim is true here and would fail if $r(t) = 12 - 3t^{2}$ were replaced by $12 - 12t^{2}$ .

Cheatsheet

Core Reference

	Antiderivative	Frequent error
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	dividing by $n$
$\frac{1}{x}$	$\ln\lvert x \rvert + C$	applying the power rule
$k$	$kx + C$	dropping the $x$
$e^{x}$	$e^{x} + C$	none
$\cos x$	$\sin x + C$	none
$\sin x$	$-\cos x + C$	losing the minus
$f(ax+b)$	$\frac{1}{a}F(ax+b) + C$	multiplying by $a$

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Unit 6Antiderivatives for Basic Functions

Concept 4.109Built for May 2027 AP Exam

Antiderivatives for Basic Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Mostly indirect use. A no-calculator part asks for a total from a polynomial rate, and the antiderivative is the first step. Occasionally an item asks directly for $\int f(x)\,dx$ and offers four candidate families.
What you’ll do: Reverse one rule at a time, check the result by differentiating it back, and keep the constant when the integral has no limits.
Memory hook: Up one, divide by the new one. The exponent rises by one and the same number lands underneath.
Use it on the exam: For MCQs: Differentiate the choices instead of integrating. Only one candidate will return the integrand, and the check takes less time than the computation. For FRQs: When the antiderivative is a step inside a larger part, differentiate your answer in the margin before you use it. One line of checking prevents an error that would otherwise cost every point in the part.

Concept Review

A Family, Not a Function

Figure 1 — Every curve in the family has the same slope at the same $x$ , so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

If $F'(x) = f(x)$ on an interval, then $F$ is an antiderivative of $f$ there, and so is $F(x) + C$ for any constant $C$ . The indefinite integral records the whole family:

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

A definite integral does not need the constant, because the two evaluations cancel it. An indefinite integral does, and dropping it is a scoring error rather than a stylistic one.

Every curve in the family has the same slope at the same $x$, so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

The Rules Worth Memorizing

Integrand	Antiderivative	Check by differentiating
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	$(n+1)\frac{x^{n}}{n+1} = x^{n}$
$x^{-1} = \frac{1}{x}$	$\ln\lvert x \rvert + C$	$\frac{1}{x}$
$k$ , a constant	$kx + C$	$k$
$e^{x}$	$e^{x} + C$	$e^{x}$
$\cos x$	$\sin x + C$	$\cos x$
$\sin x$	$-\cos x + C$	$\sin x$
$\sec^{2} x$	$\tan x + C$	$\sec^{2} x$

Linear Inner Functions

The chain rule leaves a trace that can be reversed when the inner function is linear:

$\int f(ax + b)\,dx = \frac{1}{a}F(ax + b) + C$

Functions With No Elementary Antiderivative

Checking Is a One-Line Habit

Common Misconception Check

Myth 1: “ $\int x^{-1}\,dx = \frac{x^{0}}{0}$ .” Truth: That expression is undefined. The antiderivative of $x^{-1}$ is $\ln\lvert x \rvert$ , and this is the single exception the power rule cannot cover.
Myth 2: “The $+C$ is decoration.” Truth: It is the difference between a family and a member. Any part that asks for a particular solution depends on it.
Myth 3: “Every continuous function has an antiderivative you can write down.” Truth: Every continuous function has an antiderivative. Very few have one built from elementary functions.

Topic-Specific Study Method

Reverse One Rule, Then Check

Exam disguise	What you see on the page	What your brain must translate	Required execution
“Evaluate the integral” with a polynomial	powers and coefficients	reverse power rule	raise, divide, check
A rate given as a polynomial	“how many words by the end”	antiderivative then subtract	constant term becomes $kx$
Choices that are functions	four candidate families	differentiate each	pick the one returning the integrand
$\frac{1}{x}$ in the integrand	a negative power	the logarithm case	absolute value, plus $C$
A composite like $e^{3x}$	a linear inner function	reciprocal factor	divide by the inner slope

① Raise the exponent before doing anything else. Writing the new exponent down first prevents the classic slip of dividing by the old one.

② Handle the constant term separately. A bare number integrates to that number times $x$ , not to zero and not to a higher power.

③ Differentiate the answer. If the result is not the integrand, the error is in the exponent or in a sign, and both are visible in one line.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\int\left(6x^{2} - \frac{4}{x} + 5\right)dx$ .

Standard Execution:

Power term. $\int 6x^{2}\,dx = 2x^{3}$ .

Logarithm term. $\int \frac{4}{x}\,dx = 4\ln\lvert x \rvert$ .

Constant term. $\int 5\,dx = 5x$ .

Assemble. $2x^{3} - 4\ln\lvert x \rvert + 5x + C$ .

Check. Differentiating gives $6x^{2} - \frac{4}{x} + 5$ , which matches.

Solution: An antiderivative is $F(t) = -t^{3} + 4t^{2} + 100t$ . Then $F(10) = -1000 + 400 + 1000 = 400$ and $F(0) = 0$ , so the total is $400$ words. Two checks are worth doing: the units are words because a rate in words per minute was integrated over minutes, and differentiating $F$ returns $W$ exactly, which confirms that the constant term became $100t$ rather than disappearing.

Common Traps & Failure Signals:

Dividing by the old exponent (Signal: a coefficient of $6x^{3}$ from $6x^{2}$ ): divide by the new exponent, which is the one you just wrote.
Constant term vanished (Signal: an antiderivative with no linear term when the integrand had a constant): a constant integrates to $kx$ .
Missing constant (Signal: an indefinite integral with no $+C$ ): the family needs it.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Exponent Off By One: Write the new exponent first.
Sign Dropped: You integrated $\sin x$ to $\cos x$ and lost the minus, which is not a memory lapse but a direction error, since the derivative of $\cos$ carries the minus and reversing must carry it too.
Constant Ignored: You treated a bare number as if it integrated to nothing, so the antiderivative is off by a linear term that looks harmless on a graph and changes the value of every definite integral built on it.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Two of the nine released exams since 2017 contain a part in which producing an antiderivative from the basic rules is the scored step, and both are no-calculator parts
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; The skill otherwise appears inside other parts and dominates the multiple-choice section, where the reverse power rule is tested directly
FRQ Years Appeared: 2019 (Q3b), 2025 (Q3D) ---

Practice Examples

MCQ 1

The Reverse Power Rule

$\int\left(8x^{3} - 6x\right)dx =$

MCQ 2

The Logarithm Case

$\int \frac{5}{x}\,dx =$

MCQ 3

A Composite With a Linear Inside

$\int \cos(4x)\,dx =$

FRQ 4

From a Rate to a Total

A pump moves water at a rate modeled by $r(t) = 12 - 3t^{2}$ gallons per minute for $0 \le t \le 2$ minutes.

(a) Find $\int r(t)\,dt$ .

(b) Find the total volume of water moved during the two minutes.

Cheatsheet

Core Reference

	Antiderivative	Frequent error
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	dividing by $n$
$\frac{1}{x}$	$\ln\lvert x \rvert + C$	applying the power rule
$k$	$kx + C$	dropping the $x$
$e^{x}$	$e^{x} + C$	none
$\cos x$	$\sin x + C$	none
$\sin x$	$-\cos x + C$	losing the minus
$f(ax+b)$	$\frac{1}{a}F(ax+b) + C$	multiplying by $a$

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Unit 6Antiderivatives for Basic Functions

Concept 4.109Built for May 2027 AP Exam

Antiderivatives for Basic Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Mostly indirect use. A no-calculator part asks for a total from a polynomial rate, and the antiderivative is the first step. Occasionally an item asks directly for $\int f(x)\,dx$ and offers four candidate families.
What you’ll do: Reverse one rule at a time, check the result by differentiating it back, and keep the constant when the integral has no limits.
Memory hook: Up one, divide by the new one. The exponent rises by one and the same number lands underneath.
Use it on the exam: For MCQs: Differentiate the choices instead of integrating. Only one candidate will return the integrand, and the check takes less time than the computation. For FRQs: When the antiderivative is a step inside a larger part, differentiate your answer in the margin before you use it. One line of checking prevents an error that would otherwise cost every point in the part.

Concept Review

A Family, Not a Function

Figure 1 — Every curve in the family has the same slope at the same $x$ , so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

If $F'(x) = f(x)$ on an interval, then $F$ is an antiderivative of $f$ there, and so is $F(x) + C$ for any constant $C$ . The indefinite integral records the whole family:

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

A definite integral does not need the constant, because the two evaluations cancel it. An indefinite integral does, and dropping it is a scoring error rather than a stylistic one.

Every curve in the family has the same slope at the same $x$, so all of them are antiderivatives of the same function. The constant only decides which one you are standing on.

The Rules Worth Memorizing

Integrand	Antiderivative	Check by differentiating
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	$(n+1)\frac{x^{n}}{n+1} = x^{n}$
$x^{-1} = \frac{1}{x}$	$\ln\lvert x \rvert + C$	$\frac{1}{x}$
$k$ , a constant	$kx + C$	$k$
$e^{x}$	$e^{x} + C$	$e^{x}$
$\cos x$	$\sin x + C$	$\cos x$
$\sin x$	$-\cos x + C$	$\sin x$
$\sec^{2} x$	$\tan x + C$	$\sec^{2} x$

Linear Inner Functions

The chain rule leaves a trace that can be reversed when the inner function is linear:

$\int f(ax + b)\,dx = \frac{1}{a}F(ax + b) + C$

Functions With No Elementary Antiderivative

Checking Is a One-Line Habit

Common Misconception Check

Myth 1: “ $\int x^{-1}\,dx = \frac{x^{0}}{0}$ .” Truth: That expression is undefined. The antiderivative of $x^{-1}$ is $\ln\lvert x \rvert$ , and this is the single exception the power rule cannot cover.
Myth 2: “The $+C$ is decoration.” Truth: It is the difference between a family and a member. Any part that asks for a particular solution depends on it.
Myth 3: “Every continuous function has an antiderivative you can write down.” Truth: Every continuous function has an antiderivative. Very few have one built from elementary functions.

Topic-Specific Study Method

Reverse One Rule, Then Check

Exam disguise	What you see on the page	What your brain must translate	Required execution
“Evaluate the integral” with a polynomial	powers and coefficients	reverse power rule	raise, divide, check
A rate given as a polynomial	“how many words by the end”	antiderivative then subtract	constant term becomes $kx$
Choices that are functions	four candidate families	differentiate each	pick the one returning the integrand
$\frac{1}{x}$ in the integrand	a negative power	the logarithm case	absolute value, plus $C$
A composite like $e^{3x}$	a linear inner function	reciprocal factor	divide by the inner slope

① Raise the exponent before doing anything else. Writing the new exponent down first prevents the classic slip of dividing by the old one.

② Handle the constant term separately. A bare number integrates to that number times $x$ , not to zero and not to a higher power.

③ Differentiate the answer. If the result is not the integrand, the error is in the exponent or in a sign, and both are visible in one line.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find $\int\left(6x^{2} - \frac{4}{x} + 5\right)dx$ .

Standard Execution:

Power term. $\int 6x^{2}\,dx = 2x^{3}$ .

Logarithm term. $\int \frac{4}{x}\,dx = 4\ln\lvert x \rvert$ .

Constant term. $\int 5\,dx = 5x$ .

Assemble. $2x^{3} - 4\ln\lvert x \rvert + 5x + C$ .

Check. Differentiating gives $6x^{2} - \frac{4}{x} + 5$ , which matches.

Solution: An antiderivative is $F(t) = -t^{3} + 4t^{2} + 100t$ . Then $F(10) = -1000 + 400 + 1000 = 400$ and $F(0) = 0$ , so the total is $400$ words. Two checks are worth doing: the units are words because a rate in words per minute was integrated over minutes, and differentiating $F$ returns $W$ exactly, which confirms that the constant term became $100t$ rather than disappearing.

Common Traps & Failure Signals:

Dividing by the old exponent (Signal: a coefficient of $6x^{3}$ from $6x^{2}$ ): divide by the new exponent, which is the one you just wrote.
Constant term vanished (Signal: an antiderivative with no linear term when the integrand had a constant): a constant integrates to $kx$ .
Missing constant (Signal: an indefinite integral with no $+C$ ): the family needs it.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Exponent Off By One: Write the new exponent first.
Sign Dropped: You integrated $\sin x$ to $\cos x$ and lost the minus, which is not a memory lapse but a direction error, since the derivative of $\cos$ carries the minus and reversing must carry it too.
Constant Ignored: You treated a bare number as if it integrated to nothing, so the antiderivative is off by a linear term that looks harmless on a graph and changes the value of every definite integral built on it.

Past 10-Year AP Exam Appearance

Testing Frequency: Rare in FRQs; Two of the nine released exams since 2017 contain a part in which producing an antiderivative from the basic rules is the scored step, and both are no-calculator parts
10-Year Total Count: 2 of 9 Exams · 2 FRQ Parts; The skill otherwise appears inside other parts and dominates the multiple-choice section, where the reverse power rule is tested directly
FRQ Years Appeared: 2019 (Q3b), 2025 (Q3D) ---

Practice Examples

MCQ 1

The Reverse Power Rule

$\int\left(8x^{3} - 6x\right)dx =$

MCQ 2

The Logarithm Case

$\int \frac{5}{x}\,dx =$

MCQ 3

A Composite With a Linear Inside

$\int \cos(4x)\,dx =$

FRQ 4

From a Rate to a Total

A pump moves water at a rate modeled by $r(t) = 12 - 3t^{2}$ gallons per minute for $0 \le t \le 2$ minutes.

(a) Find $\int r(t)\,dt$ .

(b) Find the total volume of water moved during the two minutes.

Cheatsheet

Core Reference

	Antiderivative	Frequent error
$x^{n}$ , $n \ne -1$	$\frac{x^{n+1}}{n+1} + C$	dividing by $n$
$\frac{1}{x}$	$\ln\lvert x \rvert + C$	applying the power rule
$k$	$kx + C$	dropping the $x$
$e^{x}$	$e^{x} + C$	none
$\cos x$	$\sin x + C$	none
$\sin x$	$-\cos x + C$	losing the minus
$f(ax+b)$	$\frac{1}{a}F(ax+b) + C$	multiplying by $a$

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