Unit 6: Antiderivatives

Antiderivatives Overview
Definite Integrals = Fundamental Theorem of Calculus Part 2
3 Topics
Riemann Sums Become Definite Integrals
The Actual Fundamental Theorem of Calculus Part 2
Net Area vs Total Area
Indefinite Integrals
The Two Main Antiderivative Rules
The Two Main Antiderivative Rules
Always Step 1: Simplify and Look for Algebraic Rewrites.
5 Topics
Roots or Radicals
x’s On the Bottom of a Fraction
Trig Functions Raised to a Power
Look to Make Your Life Easier
Check the Bounds (NEW)
Rule 1: Constants
3 Topics
Identifier: Antiderivative of Constants
Method: Antiderivative of Constants
Example 1: Antiderivative of a Constant
Rule 2: Power Rule
6 Topics
Identifier: Power Rule Antiderivative
Method: Power Rule Antiderivative
Example 1: Definite Integral Power Rule
Example 1: Indefinite Integral Power Rule
Example 2: Definite Integral Power Rule
Example 2: Indefinite Integral Power Rule
Special Case Antiderivatives (The ones you have to memorize)
Special Case Antiderivatives
Six Trig Functions
4 Topics
Identifier: Trig Function Antiderivatives
Method: Trig Function Antiderivatives
Example 1: Definite Integral Trig Function
Example 1: Indefinite Integral Trig Functions
Six Inverse Trig Functions
4 Topics
Identifier: Inverse Trig Antiderivatives
Method: Inverse Trig Antiderivatives
Example 1: Definite Integral Inverse Trig
Example 1: Indefinite Integral Inverse Trig
e ( nx )
4 Topics
Identifier: e^(x) Antiderivative
Method: e^(x) Antiderivatives
Example 1: e^(x) Definite Integral
Example 1: e^(ax) Indefinite Integral
a ( nx )
4 Topics
Identifier: a^(nx) Antiderivative
Method: a^(nx) Antiderivative
Example 1: Definite Integral Exponential Function
Example 1: Indefinite Integral of Exponential Function
n x
4 Topics
Identifier: ln(x) Antiderivative
Method: ln(x) Antiderivative
Example 1: Definite Integral ln(x)
Example 1: Indefinite Integral ln(x)
U-Substitution
U-Substitution
4 Topics
Identifier: U-Substitution
Method: U-Substitution
Example 1: Indefinite Integral U-Substitution
Example 2: Definite Integral U-Substitution
Initial Value Problems: Finding the +C
Initial Value Problems
3 Topics
Identifier: Initial Value Problem
Method: Initial Value Problem
Example 1: Initial Value Problem
Fundamental Theorem of Calculus Part 1
Fundamental Theorem of Calculus Part 1
4 Topics
Identifier: Fundamental Theorem of Calculus Pt. 1
Method: Fundamental Theorem of Calculus Pt. 1
Example 1: Fundamental Theorem of Calculus Pt. 1
Example 2: Fundamental Theorem of Calculus Pt. 1
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Look to Make Your Life Easier

Unit 6: Antiderivatives Always Step 1: Simplify and Look for Algebraic Rewrites. Look to Make Your Life Easier

Sometimes a simple algebra move at the beginning of a math problem can change the level of difficulty dramatically.

There is no set of rules here, but just reminded yourself to stop and take a look at the problem first before you go rushing into it.

Here are just a couple of examples of what I mean. You will see that there is some personal preference in these choices.

Algebra: Look to Distribute

Less parentheses are usually better than more parentheses.

Original

Rewrite

f ( x ) = tan ( 3 π ( 2 x + 3 ) )

f ( x ) = tan ( 6 π x + 9 π )

f ( x ) = 5 x ( x 3 – 9 x + 7 )

f ( x ) = 5 x 4 – 45 x 2 + 35 x

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