Reverse Product Rule Slides The Reverse Product Rule
Lesson 1: Deriving Integration by Parts
The "Failure" Hook
Try to integrate the following using u-Substitution:
\[ \int x \cos(x) \, dx \]
Wait...
If \( u = x \), then \( du = dx \). We still have \( \cos(x) \).
And...
If \( u = \cos(x) \), then \( du = -\sin(x) \, dx \). Where is our sine?
Substitution fails when parts of the product are unrelated.
Thinking Backwards
Recall the Product Rule of Differentiation
\[ \frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x) \]
To find the integral of a product, what if we integrated both sides?
Step 1: Integrate
Step 2: Isolate
Step 3: Parts Formula
The Derivation
1
\[ \int \frac{d}{dx}[uv] \, dx = \int u \, dv + \int v \, du \]
2
\[ uv = \int u \, dv + \int v \, du \]
3
\[ \int u \, dv = uv - \int v \, du \]
"The integral of u with respect to v is equal to the product uv minus the integral of v with respect to u."
Anatomy of "The Parts"
\( u \)
The part we want to differentiate. Ideally, it becomes simpler when derived.
\( dv \)
The part we want to integrate. It must be something we actually know how to integrate!
Pro Tip:
The goal is for \( \int v \, du \) to be easier than the original integral.
Formula Discovery Worksheet Parts Formula Discovery
Calculus: Integration Techniques
NAME:
DATE:
1. Thinking Backwards
Before we can integrate a product, we must recall how we differentiate a product. Write the Product Rule below:
\( \frac{d}{dx} [u \cdot v] = \) ____________________________________________________________________
2. The Derivation Path
Follow these steps to discover the Integration by Parts formula. Use the differentials \( du = u' \, dx \) and \( dv = v' \, dx \).
Step A:
Integrate both sides of the Product Rule with respect to \( x \):
\( \int \frac{d}{dx} [uv] \, dx = \int u \, v' \, dx + \int v \, u' \, dx \)
Step B:
The integral of a derivative simplifies. Rewrite the left side. Also, substitute \( dv \) for \( v' \, dx \) and \( du \) for \( u' \, dx \):
Step C:
Now, isolate the term \( \int u \, dv \) to find our final formula:
\( \int u \, dv = \) __________________________________________
3. Identifying "The Parts"
For each integral below, identify a choice for \( u \) and \( dv \). Remember: We want \( u \) to get simpler when derived and \( dv \) to be easy to integrate.
A) \( \int x \cos(x) \, dx \)
\( u = \) __________________
\( dv = \) _________________
\( du = \) _________________
\( v = \) __________________
B) \( \int x e^x \, dx \)
\( u = \) __________________
\( dv = \) _________________
\( du = \) _________________
\( v = \) __________________
4. The Hook Challenge
Using your parts from Problem A above and the formula you derived, solve the integral: \( \int x \cos(x) \, dx \)
Formula Discovery Answer Key Parts Formula Discovery ANSWER KEY
Calculus: Integration Techniques
1. Thinking Backwards
\( \frac{d}{dx} [u \cdot v] = \) \( u \cdot v' + v \cdot u' \)
2. The Derivation Path
Step A:
Integrate both sides of the Product Rule with respect to \( x \):
\( \int \frac{d}{dx} [uv] \, dx = \int u \, v' \, dx + \int v \, u' \, dx \)
Step B:
\( uv = \int u \, dv + \int v \, du \)
(Note: \( v' \, dx \) became \( dv \) and \( u' \, dx \) became \( du \))
Step C:
Isolate the term \( \int u \, dv \):
\( \int u \, dv = \) \( uv - \int v \, du \)
3. Identifying "The Parts"
A) \( \int x \cos(x) \, dx \)
\( u = \) \( x \)
\( dv = \) \( \cos(x) \, dx \)
\( du = \) \( dx \)
\( v = \) \( \sin(x) \)
B) \( \int x e^x \, dx \)
\( u = \) \( x \)
\( dv = \) \( e^x \, dx \)
\( du = \) \( dx \)
\( v = \) \( e^x \)
4. The Hook Challenge
\[ \int x \cos(x) \, dx = (x)(\sin x) - \int (\sin x) \, dx \]
\[ = x \sin x - (-\cos x) + C \]
\( = x \sin x + \cos x + C \)
Strategic Choice Slides LIATE
The Art of the Choice
Lesson 2: The LIATE Strategy
The Choice Dilemma
Consider: \[ \int x \ln(x) \, dx \]
WRONG CHOICE
Let \( u = x \), \( dv = \ln(x) \, dx \)
Problem:
Do you know the integral of \( \ln(x) \)? Not easily!
SMART CHOICE
Let \( u = \ln(x) \), \( dv = x \, dx \)
Benefit:
The derivative of \( \ln(x) \) is \( 1/x \), which is much simpler!
The LIATE Hierarchy
Pick your u based on which function type appears first in this list:
L
Logarithmic
\( \ln(x) \)
I
Inverse Trig
\( \arctan(x) \)
A
Algebraic
\( x^2, 3x^4 \)
T
Trig
\( \sin(x), \cos(x) \)
E
Exponential
\( e^x, 2^x \)
Rule of Thumb: Choose the one that is "hardest to integrate" as u.
Practice Picking
1. \( \int x^2 \sin(x) \, dx \)
A vs T
u = x²
2. \( \int x \arctan(x) \, dx \)
A vs I
u = arctan(x)
3. \( \int e^x \cos(x) \, dx \)
E vs T
u = cos(x)*
*Note: For T vs E, either can technically work, but LIATE suggests Trig first!
LIATE Sorting Activity The LIATE Choice Maze
Calculus: Strategy Activity
Instructions
For each integral below, identify the two types of functions present using LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential). Then, circle the function that should be chosen as \( u \) .
\( \int x^3 \ln(x) \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int x^2 e^{5x} \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int \arcsin(x) \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int x \cos(3x) \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int \frac{\ln(x)}{x^2} \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int e^x \sin(x) \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int x^n \sin(ax) \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int \frac{\arctan(x)}{1} \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
\( \int 5x^4 e^x \, dx \)
Types Present:
L I A T E
L I A T E
Choice:
\( u = \) _________________
"L-I-A-T-E: The higher it is on the list, the more it wants to be 'u'!"
Strategic Selection Worksheet Strategic Selection Practice
Calculus: Integration by Parts
NAME:
1. Rapid Choice (LIATE)
For each integral, identify the function types (L, I, A, T, E) and select \( u \).
\( \int x^4 \ln(x) \, dx \) \( u = \) __________
\( \int x e^{-x} \, dx \) \( u = \) __________
\( \int \arctan(2x) \, dx \) \( u = \) __________
\( \int x \sec^2(x) \, dx \) \( u = \) __________
2. Full Integration Practice
A) Evaluate \( \int x^2 \ln(x) \, dx \)
\( u = \) __________________
\( dv = \) _________________
\( du = \) _________________
\( v = \) __________________
B) Evaluate \( \int (3x - 1) e^x \, dx \)
\( u = \) __________________
\( dv = \) _________________
\( du = \) _________________
\( v = \) __________________
3. The Choice Trap
Evaluate the following. Should you use Substitution or Integration by Parts ? Explain your choice first.
\( \int x \cos(x^2) \, dx \)
Strategy & Explanation:
Solution:
Strategic Selection Answer Key Strategic Selection ANSWER KEY
Calculus: Integration by Parts
1. Rapid Choice (LIATE)
\( \int x^4 \ln(x) \, dx \) \( u = \ln(x) \) (L)
\( \int x e^{-x} \, dx \) \( u = x \) (A)
\( \int \arctan(2x) \, dx \) \( u = \arctan(2x) \) (I)
\( \int x \sec^2(x) \, dx \) \( u = x \) (A)
2. Full Integration Practice
A) \( \int x^2 \ln(x) \, dx \)
\( u = \ln(x) \)
\( dv = x^2 \, dx \)
\( du = \frac{1}{x} \, dx \)
\( v = \frac{1}{3}x^3 \)
\( uv - \int v \, du = \frac{1}{3}x^3 \ln(x) - \int \frac{1}{3}x^3 \cdot \frac{1}{x} \, dx \)
B) \( \int (3x - 1) e^x \, dx \)
\( u = 3x - 1 \)
\( dv = e^x \, dx \)
\( du = 3 \, dx \)
\( v = e^x \)
\( (3x-1)e^x - \int 3e^x \, dx = \mathbf{(3x-1)e^x - 3e^x + C} \)
3. The Choice Trap
\( \int x \cos(x^2) \, dx \)
Strategy & Explanation:
Use Substitution.
The derivative of the "inner" function \( x^2 \) is \( 2x \). Since an \( x \) term is already present outside, substitution is much simpler than parts.
Solution:
Let \( w = x^2 \), then \( dw = 2x \, dx \). The integral becomes:
\( \frac{1}{2} \int \cos(w) \, dw = \frac{1}{2} \sin(w) + C = \mathbf{\frac{1}{2} \sin(x^2) + C} \)
The Tabular Method Slides The Tabular Hack
Lesson 3: Repeated Integration Made Easy
The Efficiency Crisis
Try integrating: \[ \int x^4 e^x \, dx \]
Standard Method:
Parts Iteration 1 (\( u = x^4 \))
Parts Iteration 2 (\( u = x^3 \))
Parts Iteration 3 (\( u = x^2 \))
Parts Iteration 4 (\( u = x \))
There must be a faster way... and there is.
The Tabular Method
Works perfectly for: Polynomial × (Trig or Exponential)
Sign
D (Derive u)
I (Integrate dv)
\( x^2 \)
\( \sin(x) \)
-
\( 2x \)
\( -\cos(x) \)
\( 2 \)
\( -\sin(x) \)
-
\( 0 \)
\( \cos(x) \)
Connecting the Dots
Diagonal Multiplication!
Row 1 (+) \(\times\) Row 2 (I)
Row 2 (-) \(\times\) Row 3 (I)
Row 3 (+) \(\times\) Row 4 (I)
Stop when the derivative side reaches zero . Your final answer is simply the sum of these diagonal products!
Repeated Integration Worksheet Tabular Mastery
Calculus: Efficient Integration
UNIT: PARTS
LESSON 3
1. Method Comparison
Evaluate \( \int x^2 e^x \, dx \) using both methods to compare efficiency.
Standard Parts (Long Way)
Tabular Method (Efficient)
Sign
D (u)
I (dv)
Final Answer Area:
2. Tabular Method Practice
A) \( \int x^3 \cos(x) \, dx \)
Differentiate u until 0.
B) \( \int x^4 e^{2x} \, dx \)
Watch your chain rule on integration!
3. Know the Limits
Explain why the Tabular Method is not a good choice for evaluating the following integral:
\( \int x^2 \ln(x) \, dx \)
Repeated Integration Answer Key Tabular Mastery ANSWER KEY
Calculus: Efficient Integration
1. Method Comparison: \( \int x^2 e^x \, dx \)
Standard Method Summary:
1. \( u=x^2, dv=e^x \to x^2e^x - \int 2xe^x \, dx \)
2. \( u=2x, dv=e^x \to x^2e^x - [2xe^x - \int 2e^x \, dx] \)
3. \( x^2e^x - 2xe^x + 2e^x + C \)
SignDI +\( x^2 \)\( e^x \) -\( 2x \)\( e^x \) +\( 2 \)\( e^x \) -\( 0 \)\( e^x \)
\( e^x(x^2 - 2x + 2) + C \)
2. Tabular Method Practice
A) \( \int x^3 \cos(x) \, dx \)
SDI +x³cos x -3x²sin x +6x-cos x -6-sin x +0cos x
\( = x^3 \sin x + 3x^2 \cos x - 6x \sin x - 6 \cos x + C \)
B) \( \int x^4 e^{2x} \, dx \)
SDI +x⁴e^{2x} -4x³(1/2)e^{2x} +12x²(1/4)e^{2x} -24x(1/8)e^{2x} +24(1/16)e^{2x} -0(1/32)e^{2x}
\( = \frac{1}{2}x^4 e^{2x} - x^3 e^{2x} + 3x^2 e^{2x} - 3x e^{2x} + \frac{3}{2}e^{2x} + C \)
3. Know the Limits
Why Tabular Fails for \( \int x^2 \ln(x) \, dx \):
According to LIATE, we must pick \( u = \ln(x) \). The derivatives of \( \ln(x) \) are \( 1/x, -1/x^2, 2/x^3 \), and so on. These never reach zero , which is the required "stopping point" for the standard tabular method. Standard parts or a "modified" tabular method (integrating across a row) is needed instead.
Breaking the Loop Slides Breaking the Loop
Lesson 4: Cyclic Integrals
The Infinite Loop
What happens with \[ \int e^x \cos(x) \, dx \]?
Round 1:
\( u = \cos(x), dv = e^x \, dx \)
\( \to e^x \cos(x) + \int e^x \sin(x) \, dx \)
Round 2:
\( u = \sin(x), dv = e^x \, dx \)
\( \to e^x \sin(x) - \int e^x \cos(x) \, dx \)
WE ARE BACK TO THE START!
The Algebraic Escape Hatch
Stop integrating and start solving for I.
Let \( I = \int e^x \cos(x) \, dx \)
Our work showed:
\( 2I = e^x \cos(x) + e^x \sin(x) \)
The Escape hatch (Step 3)
Divide by the coefficient to isolate \( I \):
\( I = \frac{e^x \cos(x) + e^x \sin(x)}{2} + C \)
This works whenever repeated parts returns the exact same integral (with a different sign or coefficient).
Cyclic Integrals Worksheet Cyclic Loop Challenges
Calculus: Advanced Parts
NAME:
1. Spotting the Cycle
Not every integral is cyclic. Check the box if you think the integral will require the "Algebraic Loop" method.
\( \int x^2 \sin(x) \, dx \)
\( \int e^x \cos(2x) \, dx \)
\( \int \sin(x) \cos(x) \, dx \)
\( \int e^{-x} \sin(x) \, dx \)
2. The Scaffolded Loop
Solve \( I = \int e^x \sin(x) \, dx \) by following the steps.
Step A: First Application of Parts
\( u = \sin(x) \), \( du = \) ________
\( dv = e^x \, dx \), \( v = \) ________
Result: \( I = \) __________________________________________________
Step B: Second Application (of the new integral)
\( u = \cos(x) \), \( du = \) ________
\( dv = e^x \, dx \), \( v = \) ________
Result: \( I = (uv_1) - [ (uv_2) - \int v_2 du_2 ] \)
Step C: Solve for I
Identify where the original \( I \) has reappeared and solve the equation.
3. Final Challenge
Evaluate: \( \int e^{2x} \cos(3x) \, dx \)
Cyclic Integrals Answer Key Cyclic Loop Challenges ANSWER KEY
Calculus: Advanced Parts
1. Spotting the Cycle
\( \int x^2 \sin(x) \, dx \)
\( \int e^x \cos(2x) \, dx \)
\( \int \sin(x) \cos(x) \, dx \)
\( \int e^{-x} \sin(x) \, dx \)
2. The Scaffolded Loop: \( I = \int e^x \sin(x) \, dx \)
Step A: First Application
\( u = \sin x, du = \cos x \, dx \); \( dv = e^x \, dx, v = e^x \)
\( I = e^x \sin x - \int e^x \cos x \, dx \)
Step B: Second Application
\( u = \cos x, du = -\sin x \, dx \); \( dv = e^x \, dx, v = e^x \)
\( I = e^x \sin x - [ e^x \cos x - \int e^x (-\sin x) \, dx ] \)
\( I = e^x \sin x - e^x \cos x - \int e^x \sin x \, dx \)
Step C: Solve for I
\( I = e^x \sin x - e^x \cos x - I \)
\( 2I = e^x(\sin x - \cos x) \)
\( I = \frac{e^x(\sin x - \cos x)}{2} + C \)
3. Final Challenge Answer
1st Parts: u = cos(3x), dv = e^(2x) dx → du = -3sin(3x) dx, v = (1/2)e^(2x)
I = (1/2)e^(2x)cos(3x) + (3/2) ∫ e^(2x)sin(3x) dx
2nd Parts: u = sin(3x), dv = e^(2x) dx → du = 3cos(3x) dx, v = (1/2)e^(2x)
I = (1/2)e^(2x)cos(3x) + (3/2) [ (1/2)e^(2x)sin(3x) - (3/2) ∫ e^(2x)cos(3x) dx ]
I = (1/2)e^(2x)cos(3x) + (3/4)e^(2x)sin(3x) - (9/4)I
(13/4)I = (2/4)e^(2x)cos(3x) + (3/4)e^(2x)sin(3x)
I = \frac{1}{13} e^{2x} [2 \cos(3x) + 3 \sin(3x)] + C
Definite Parts Slides Closing the Bounds
Lesson 5: Definite Integrals by Parts
The Definite Formula
When limits are involved, the formula must respect the Fundamental Theorem of Calculus :
\[ \int_a^b u \, dv = [u \cdot v]_a^b - \int_a^b v \, du \]
Crucial Point 1
The \( uv \) term is evaluated at both \( a \) and \( b \).
Crucial Point 2
The integral term is evaluated after the second integration.
Application: Center of Mass
Finding the center of mass \( \bar{x} \) of a 1-meter rod with density \( \rho(x) = x e^x \) requires:
\[ \text{Moment} = \int_0^1 x^2 e^x \, dx \]
This is a definite integral that requires the Tabular Method!
Pivot Point
Final Notational Tips
Bracket Notation
Use \( [uv]_a^b \) clearly to show that you haven't forgotten to plug in the values yet.
Zeroes are Friends
Definite integrals by parts often involve \( x^n \) or \( \sin(x) \). Evaluating at 0 often simplifies the expression significantly!
One Shot Evaluation
You can also wait and evaluate the entire antiderivative at the very end. Just be careful with signs!
Integration Mastery Worksheet Integration Mastery
Calculus: Final Sequence Practice
NAME:
1. Definite Integration
Evaluate the following definite integrals. Show your variable selection and evaluation steps.
A) \( \int_0^\pi x \sin(x) \, dx \)
Identify u, dv, du, v here...
Evaluate [uv] - integral here...
B) \( \int_1^e x^2 \ln(x) \, dx \)
2. The Variable Density Rod
A rod of length 1 meter (from \( x=0 \) to \( x=1 \)) has a non-uniform density given by \( \rho(x) = x e^x \). To find the "Moment" of the rod, you must evaluate:
\( M = \int_0^1 x(x e^x) \, dx = \int_0^1 x^2 e^x \, dx \)
Solve using the Tabular Method:
Final Evaluation:
Result should be in terms of \( e \).
3. Strategy Match-Up
Match the integral type to its most efficient strategy.
\( \int x^5 e^x \, dx \) A. Basic Parts
\( \int e^x \sin(x) \, dx \) B. LIATE Priority
\( \int x \ln(x) \, dx \) C. Tabular Method
\( \int x \cos(x) \, dx \) D. Cyclic/Loop Method
Integration Mastery Answer Key Integration Mastery ANSWER KEY
Calculus: Final Sequence Practice
1. Definite Integration
A) \( \int_0^\pi x \sin(x) \, dx \)
\( u=x, du=dx; dv=\sin x, v=-\cos x \)
\( = [-x \cos x]_0^\pi - \int_0^\pi -\cos x \, dx = [-x \cos x + \sin x]_0^\pi \)
\( = (-\pi \cos \pi + \sin \pi) - (0 + \sin 0) = \pi(1) + 0 - 0 = \mathbf{\pi} \)
B) \( \int_1^e x^2 \ln(x) \, dx \)
\( u=\ln x, du=1/x; dv=x^2, v=x^3/3 \)
\( = [\frac{1}{3}x^3 \ln x]_1^e - \int_1^e \frac{1}{3}x^2 \, dx = [\frac{1}{3}x^3 \ln x - \frac{1}{9}x^3]_1^e \)
\( = (\frac{1}{3}e^3 - \frac{1}{9}e^3) - (0 - \frac{1}{9}) = \frac{2}{9}e^3 + \frac{1}{9} = \mathbf{\frac{2e^3+1}{9}} \)
2. The Variable Density Rod
Tabular Setup for \( \int_0^1 x^2 e^x \, dx \):
(+) x² | e^x
(-) 2x | e^x
(+) 2 | e^x
(-) 0 | e^x
Antiderivative: \( e^x(x^2 - 2x + 2) \)
Evaluate from 0 to 1:
\( e^1(1 - 2 + 2) - e^0(0 - 0 + 2) \)
\( = e(1) - 1(2) = \mathbf{e - 2} \)
3. Strategy Match-Up Results
\( \int x^5 e^x \, dx \) → C. Tabular
\( \int e^x \sin(x) \, dx \) → D. Cyclic
\( \int x \ln(x) \, dx \) → B. LIATE
\( \int x \cos(x) \, dx \) → A. Basic
End of Sequence: Integration by Parts Mastery.
