🎯⭐ INTERACTIVE LESSON

Antiderivatives & Indefinite Integrals

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Antiderivatives & Indefinite Integrals - Complete Interactive Lesson

Part 1: What is an Antiderivative?

Antiderivatives & Indefinite Integrals

Part 1 of 7 — What is an Antiderivative?

Definition

An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$ .

The indefinite integral represents the family of all antiderivatives:

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

The " $+ C$ " is essential! Since the derivative of a constant is 0, there are infinitely many antiderivatives.

Power Rule for Integration

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

This reverses the power rule for differentiation.

Worked Examples

$f(x)$	$\int f(x)\,dx$
$x^4$

Power Rule for Integration 🎯

Key Takeaways — Part 1

An antiderivative reverses differentiation: if $F' = f$ , then $\int f = F + C$
Power Rule: add 1 to the exponent, divide by the new exponent
for indefinite integrals

Part 2: Essential Antiderivative Formulas

Antiderivatives

Part 2 of 7 — Essential Antiderivative Formulas

Complete Table of Basic Antiderivatives

Function	Antiderivative
$x^n$ $(n \neq -1)$

Part 3: Trig Antiderivatives

Antiderivatives

Part 3 of 7 — Initial Value Problems (IVPs)

Finding Specific Antiderivatives

An initial condition pins down the value of $C$ :

Given: $f'(x) = 3x^2 - 4x + 1$ and . Find .

Part 4: Rewriting Before Integrating

Antiderivatives

Part 4 of 7 — Rewriting Before Integrating

Algebraic Manipulation

Many integrals require rewriting before applying basic rules.

Expand Products

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

Part 5: Inverse Trig Antiderivatives

Antiderivatives

Part 5 of 7 — Inverse Trig Antiderivatives

Three Key Formulas

$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C$

Part 6: Mixed Practice

Antiderivatives

Part 6 of 7 — Mixed Practice

Time to combine all antiderivative techniques.

Mixed Antiderivative Problems 🎯

Workshop Complete!

Practice makes perfect with antiderivatives. Always verify by differentiating your answer.

Part 7: Comprehensive Assessment

Antiderivatives — Review

Part 7 of 7 — Comprehensive Assessment

Final Assessment 🎯

Antiderivatives — Complete! ✅

You have mastered:

✅ Power Rule for integration
✅ All basic antiderivative formulas
✅ Initial Value Problems
✅ Rewriting before integrating
✅ Inverse trig antiderivatives

Always include $+C$

The special case

n = -1

\int x^{-1}\,dx = \ln|x| + C

(next part)

Memorize this table! These are the building blocks of all integration.

Essential Antiderivatives 🎯

Key Takeaways — Part 2

Memorize the complete table of basic antiderivatives
$\int \frac{1}{x}\,dx = \ln|x| + C$ (absolute value matters!)
$\int a^x\,dx = \frac{a^x}{\ln a} + C$ (not the power rule!)
Linearity: $\int [af + bg]\,dx = a\int f\,dx + b\int g\,dx$

Step 1: Find the general antiderivative. $f(x) = x^3 - 2x^2 + x + C$

Step 2: Use the initial condition to find $C$ . $f(0) = 0 - 0 + 0 + C = 5 \implies C = 5$

Answer: $f(x) = x^3 - 2x^2 + x + 5$

Position-Velocity-Acceleration

If $a(t) =$ acceleration, then:

$v(t) = \int a(t)\,dt$ (velocity)
$s(t) = \int v(t)\,dt$ (position)

Each integration introduces a constant determined by initial conditions.

Initial Value Problems 🎯

Key Takeaways — Part 3

An IVP consists of a derivative equation plus initial condition(s)
Find the general antiderivative, then plug in the initial condition to find $C$
For particle motion: integrate $a(t)$ to get $v(t)$ , integrate $v(t)$ to get $s(t)$

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

$\int \frac{3}{\sqrt{x}}\,dx = \int 3x^{-1/2}\,dx = 3 \cdot \frac{x^{1/2}}{1/2} + C = 6\sqrt{x} + C$

Simplify Then Integrate 🎯

Key Takeaways — Part 4

Expand products before integrating
Split fractions into separate terms when possible
Rewrite radicals using fractional exponents
These techniques reduce complex integrals to sums of power rule applications

$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$

$\int \frac{1}{x\sqrt{x^2 - a^2}}\,dx = \frac{1}{a}\text{arcsec}\left(\frac{|x|}{a}\right) + C$

The AP Exam tests whether you can recognize these forms:

Square root of $a^2 - x^2$ in denominator → $\arcsin$
Sum of squares $a^2 + x^2$ in denominator → $\arctan$

Inverse Trig Integrals 🎯

Key Takeaways — Part 5

$\sqrt{a^2 - x^2}$ in denominator → $\arcsin(x/a)$
$a^2 + x^2$ in denominator → $\frac{1}{a}\arctan(x/a)$
Often need u-sub to get into the standard form first

Antiderivatives & Indefinite Integrals

Key Takeaways — Part 2

Position-Velocity-Acceleration

Key Takeaways — Part 3

Split Fractions

Rewrite Radicals

Key Takeaways — Part 4

Recognition is Key

Key Takeaways — Part 5