Integration Made Simple: From Indefinite to Definite Integrals

2026.04.14

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Integration is the reverse of differentiation. It’s one of the two pillars of calculus and a must-know for AP Calculus, A-levels, and beyond.

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Key Point: Integration appears in 2–4 questions on the AP Calculus AB exam every year. Mastering the basics here sets you up for area, volume, and accumulation problems.

What Is Integration?

Definition

Integration finds a function $F(x)$ whose derivative equals $f(x)$ .

F'(x) = f(x) \quad \Rightarrow \quad \int f(x)\,dx = F(x) + C

The $C$ is the constant of integration. Since differentiation erases constants, we add it back when integrating.

Indefinite vs Definite Integrals

Type	Indefinite	Definite
Notation	$\int f(x)\,dx$	$\int_a^b f(x)\,dx$
Result	Function + C	Number (exact value)
Meaning	Find antiderivative	Calculate area

Essential Integration Formulas

Function	Integral
$x^n$ ( $n \neq -1$ )	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln\|x\| + C$
$e^x$	$e^x + C$
$\cos x$	$\sin x + C$
$\sin x$	$-\cos x + C$
$\sec^2 x$	$\tan x + C$

How to Integrate

Polynomial Integration

Integrate each term separately using the power rule.

\int (3x^2 - 4x + 1)\,dx = x^3 - 2x^2 + x + C

U-Substitution

For composite functions, substitute the inner function as $u$ .

\int 2x(x^2+1)^3\,dx

Let $u = x^2 + 1$ , $du = 2x\,dx$ :

\int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2+1)^4}{4} + C

Evaluating Definite Integrals

Find the antiderivative, then compute upper limit minus lower limit:

\int_1^3 2x\,dx = \left[x^2\right]_1^3 = 9 - 1 = 8

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Watch Out: For indefinite integrals, never forget the constant

C

. It’s a guaranteed point deduction on exams. Definite integrals don’t need

C

because it cancels out.

Applications of Integration

Area Under a Curve

The area between $y = f(x)$ and the x-axis from $a$ to $b$ :

A = \int_a^b |f(x)|\,dx

Use absolute value because regions below the x-axis give negative values.

Area Between Two Curves

A = \int_a^b |f(x) - g(x)|\,dx

Worked Examples

Example 1: Indefinite Integral

Find $\int (4x^3 - 6x + 2)\,dx$ .

Show Solution

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

Example 2: Definite Integral

Evaluate $\int_0^2 (3x^2 + 1)\,dx$ .

Show Solution

\left[x^3 + x\right]_0^2 = (8 + 2) - (0 + 0) = 10

Example 3: Area Problem

Find the area between $y = x^2$ and the x-axis from $x = 0$ to $x = 3$ .

Show Solution

Since $x^2 \geq 0$ on $[0, 3]$ , no absolute value needed:

A = \int_0^3 x^2\,dx = \left[\frac{x^3}{3}\right]_0^3 = 9

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Pro Tip: Always sketch the graph before solving area problems. It helps you identify regions above and below the x-axis, so you know where to split the integral.

Top 3 Common Mistakes

Forgetting +C in indefinite integrals — automatic point loss on every exam
Ignoring absolute value for area — regions below the x-axis give negative integrals, which cancel positive areas
Forgetting to back-substitute after u-sub — your final answer must be in terms of $x$ , not $u$

Integration Made Simple: From Indefinite to Definite Integrals

What Is Integration?

Definition

Indefinite vs Definite Integrals

Essential Integration Formulas

How to Integrate

Polynomial Integration

U-Substitution

Evaluating Definite Integrals

Applications of Integration

Area Under a Curve

Area Between Two Curves

Worked Examples

Example 1: Indefinite Integral

Example 2: Definite Integral

Example 3: Area Problem

Top 3 Common Mistakes

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