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

Power Rule for Integration 🎯

Key Takeaways — Part 1

1. An antiderivative reverses differentiation: if $F' = f$, then $\int f = F + C$
2. Power Rule: add 1 to the exponent, divide by the new exponent
3. Always include $+C$ for indefinite integrals
4. The special case $n = -1$: $\int x^{-1}\,dx = \ln|x| + C$ (next part)

Antiderivatives

Part 2 of 7 — Essential Antiderivative Formulas

Complete Table of Basic Antiderivatives

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

Essential Antiderivatives 🎯

Key Takeaways — Part 2

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

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 $f(0) = 5$. Find $f(x)$.

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

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

Antiderivatives

Part 4 of 7 — Rewriting Before Integrating

Algebraic Manipulation

Many integrals require rewriting before applying basic rules.

Expand Products

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

Split Fractions

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

Rewrite Radicals

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

Simplify Then Integrate 🎯

Key Takeaways — Part 4

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

Antiderivatives

Part 5 of 7 — Inverse Trig Antiderivatives

Three Key Formulas

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

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

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

Recognition is Key

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

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

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.

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