AP Calculus AB: Integration

Integration is one of the central ideas in AP Calculus AB because it connects algebraic functions to accumulated change. Where differentiation tells you how fast something is changing at a specific moment, integration tells you how much change builds up over an interval. In practice, integration appears in problems about area, total distance, net change, and quantities accumulated from a rate.

This article focuses on the core AB integration toolkit: antiderivatives, definite integrals, the Fundamental Theorem of Calculus (FTC), and $u$-substitution.

Antiderivatives: Reversing Differentiation

An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F^{\prime }(x)=f(x)$. The collection of all antiderivatives is written as an indefinite integral:

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

The constant $C$ (the constant of integration) matters because differentiation eliminates constants. For example, if $F(x)=x^2$, then $F^{\prime }(x)=2x$. But $x^2+7$ and $x^2-100$ also differentiate to $2x$. So:

$\int 2xdx=x^2+C$

Common antiderivative rules (AB essentials)

These are the patterns you use constantly:

• Power rule (for __MATH_INLINE_12__):

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

• Constant multiple rule:

$\int kf(x)dx=k\int f(x)dx$

• Sum rule:

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

• Special case __MATH_INLINE_16__ (log):

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

These rules allow quick integration of polynomials and simple rational expressions. For instance:

$\int (3x^2-4x+6)dx=x^3-2x^2+6x+C$

Initial value problems and accumulated constants

On AP Calculus AB, you often determine $C$ using an initial condition, such as $F(2)=5$. For example, if $F^{\prime }(x)=6x$ and $F(2)=5$:

1. Find an antiderivative: $F(x)=3x^2+C$
2. Apply the condition: $5=3(2^2)+C=12+C\Rightarrow C=-7$
3. Final function: $F(x)=3x^2-7$

This is a typical “recover the function from its derivative” problem.

Definite Integrals: Accumulation Over an Interval

A definite integral measures the net accumulation of a quantity from $a$ to $b$:

${\int }_a^{b}f(x)dx$

Geometrically, it can be interpreted as signed area between the graph of $f(x)$ and the $x$-axis over $[a,b]$:

• Areas above the $x$-axis contribute positively.
• Areas below the $x$-axis contribute negatively.

This sign convention is not a technicality. It is what makes integrals match real “net change” situations: gains minus losses, inflow minus outflow, velocity forward minus velocity backward.

Properties that matter on exams

Several properties come up repeatedly:

• Reversing limits changes the sign:

${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$

• Splitting an interval:

${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$

• If $f(x)\ge 0$ on $[a,b]$, then ${\int }_a^{b}f(x)dx\ge 0$

These let you combine integrals or reason about them without computing.

The Fundamental Theorem of Calculus (FTC)

The FTC is the key link between derivatives and integrals. In AP Calculus AB, you use two main versions.

FTC Part 1: Accumulation functions and derivatives

If $g(x)={\int }_a^{x}f(t)dt$, then (under standard continuity conditions):

$g^{\prime }(x)=f(x)$

In words: the derivative of the accumulated area up to $x$ is the original function value at $x$.

This is why integration and differentiation are “inverse processes” conceptually. It also creates a powerful way to differentiate functions defined by integrals.

When the upper limit is not just $x$

If $g(x)={\int }_a^{h(x)}f(t)dt$, then by the chain rule:

$g^{\prime }(x)=f(h(x))\cdot h^{\prime }(x)$

This is a frequent AB skill. You identify the “inside function” $h(x)$ and multiply by its derivative.

FTC Part 2: Evaluating definite integrals with antiderivatives

If $F^{\prime }(x)=f(x)$, then:

${\int }_a^{b}f(x)dx=F(b)-F(a)$

This turns the definite integral into an antiderivative evaluation problem. For example:

${\int }_1^{3}2xdx={\left[x^2\right]}_1^3=9-1=8$

The notation ${\left[F(x)\right]}_a^b$ means “evaluate at $b$ minus evaluate at $a$.”

Net change and real interpretation

FTC Part 2 supports the Net Change Theorem idea used throughout AB applications:

If a quantity changes at rate $R(t)$, then total change from $t=a$ to $t=b$ is:

${\int }_a^{b}R(t)dt$

So if $v(t)$ is velocity, then:

• Displacement: ${\int }_a^{b}v(t)dt$ (signed)
• Total distance traveled: ${\int }_a^b|v(t)|dt$ (always nonnegative)

Knowing when to use $v(t)$ versus $|v(t)|$ is a common conceptual checkpoint in AP problems.

$u$-Substitution: Integrating by Reversing the Chain Rule

$u$-substitution is the primary integration technique in Calculus AB beyond basic antiderivatives. It is essentially the chain rule in reverse.

The core pattern

If you see something like:

$\int f(g(x))g^{\prime }(x)dx$

then let $u=g(x)$, so $du=g^{\prime }(x)dx$, and the integral becomes:

$\int f(u)du$

This works best when the integrand contains a recognizable “inside function” and (up to a constant factor) its derivative.

Practical example structure (what you look for)

Common AB-friendly forms include:

• $\int (ax+b{)}^{n}dx$
• $\int \frac{1}{ax+b}dx$
• $\int \sin (ax+b)dx$ or $\int \cos (ax+b)dx$
• $\int e^{ax+b}dx$

For instance, $\int (3x+1{)}^{5}dx$ suggests $u=3x+1$ because the derivative is $3$, a constant multiple that is easy to adjust for. The goal is to rewrite everything in terms of $u$ and $du$, integrate, then substitute back to $x$.

Definite integrals and substitution limits

When using $u$-substitution in a definite integral, you can handle limits in two standard ways:

1. Convert the bounds to $u$-values and evaluate entirely in $u$.
2. Substitute back to $x$ before evaluating with the original bounds.

Both are valid. Converting bounds is often cleaner and reduces algebra mistakes, especially when the substituted expression is messy.

Common Integration Pitfalls to Avoid

Even strong students miss points due to predictable errors:

• Forgetting the constant __MATH_INLINE_83__ on indefinite integrals.
• Misusing the power rule for $n=-1$; $\int x^{-1}dx$ is $\ln |x|+C$, not $\frac{x^0}{0}$.
• Confusing area with signed integral; the integral gives net area, not total area, unless the function is nonnegative.
• Dropping the chain rule factor in FTC Part 1 when the upper limit is $h(x)$, not $x$.
• Using __MATH_INLINE_90__-substitution without matching __MATH_INLINE_91__; if the derivative of the inside function is not present (even up to a constant), substitution may not be appropriate.

How the AB Integration Pieces Fit Together

AP Calculus AB integration is less about memorizing tricks and more about recognizing a small set of powerful ideas:

• Antiderivatives undo differentiation, with $C$ capturing a family of functions.
• Definite integrals represent accumulated change and signed area.
• The Fundamental Theorem of Calculus connects both worlds, turning accumulation into derivatives (FTC 1) and integrals into antiderivative evaluations (FTC 2).
• $u$-substitution extends the basic antiderivative toolkit by reversing the chain rule.

Mastering these foundations makes later applications feel natural