Wave Power Slides Wave Power
Integrating Sine & Cosine Powers
Total Energy of a Sound Wave
Sound waves rarely look like pure sine functions. When multiple waves interfere, we get complex power-products.
The Challenge:
Find the area under the curve (total energy) of: \[ f(x) = \sin^3(x) \cos^2(x) \]
Complex Waveform Visualization
The Toolkit: Trigonometric Identities
1 Pythagorean Identity
\[ \sin^2 x + \cos^2 x = 1 \]
Used when we have an ODD power to peel off.
2 Half-Angle Identities
\[ \sin^2 x = \frac{1 - \cos 2x}{2} \] \[ \cos^2 x = \frac{1 + \cos 2x}{2} \]
Used when ALL powers are EVEN.
Scenario A
The Power of "Odd One Out"
If the power of sine OR cosine is odd, follow these steps:
1
Save one: Peel off a single factor of the odd power (this becomes your \(du\)).
2
Convert the rest: Use \( \sin^2 x + \cos^2 x = 1 \) to turn the remaining even power into the other trig function.
3
Substitute: Use u-substitution with the "base" function.
Example: For \( \int \sin^3 x \cos^2 x dx \), save one \( \sin x \). Let \( u = \cos x \).
Scenario B
When Everyone is Even
If both powers are even (including if one is zero), there is no "easy" \(du\) to save.
You must use Power-Reduction Formulas repeatedly:
\[ \cos^2 x = \frac{1}{2}(1 + \cos 2x) \]
\[ \sin^2 x = \frac{1}{2}(1 - \cos 2x) \]
Caution: This often leads to messy algebra. Stay organized!
Wave Energy Worksheet Wave Energy Worksheet
Topic: Integrating Powers of Sine and Cosine
Name: ________________________
Date: ________________________
Objective
Apply trigonometric identities and u-substitution strategies to calculate the integrals of various power combinations of \(\sin(x)\) and \(\cos(x)\). Show all steps of your substitutions.
Recall
\(\sin^2 x + \cos^2 x = 1\)
\(\sin^2 x = \frac{1}{2}(1 - \cos 2x)\)
\(\cos^2 x = \frac{1}{2}(1 + \cos 2x)\)
Integrate the "Odd Power" case: \[ \int \sin^3(x) \cos^4(x) \, dx \]
Evaluate the definite integral representing energy interference: \[ \int_{0}^{\pi/2} \cos^3(x) \, dx \]
Tackle the "All Even" case. Simplify as much as possible: \[ \int \sin^2(x) \cos^2(x) \, dx \]
Sound Engineering Challenge
A complex acoustic signal is modeled by the function \( I(t) = \sin^5(t) \). Find the average intensity over the interval \([0, \pi]\) using the formula \( \text{Avg} = \frac{1}{b-a} \int_{a}^{b} f(t) \, dt \).
Tangent and Secant Slides Tangent & Secant
Mastering the Derivative Duo
Lesson 2
tan(x) sec²(x)
The Mercator Map Mystery
In 1569, Gerardus Mercator created a map that let sailors steer constant compass courses.
The math behind stretching the latitudes required calculating: \[ \int \sec(x) \, dx \]
Surprisingly, this integral remained an unsolved mystery for nearly 100 years after the map was published!
How do we solve it?
The "Multiplication by 1" Trick
Identify the Partners
Pair A
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]
STRATEGY:
If the power of secant is even, save a \sec^2 x for \(du\).
Pair B
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]
STRATEGY:
If the power of tangent is odd, save a \sec x \tan x for \(du\).
Workflow: \(\int \sec^4(x) \, dx\)
1
Split: \[ \int (\sec^2 x) \cdot (\sec^2 x) \, dx \]
2
Convert: Replace one \(\sec^2 x\) with \( (1 + \tan^2 x) \).
3
U-Sub: Let \( u = \tan x \), then \( du = \sec^2 x \, dx \).
\[ \int (1 + u^2) \, du = u + \frac{1}{3}u^3 + C \]
The Blueprint Reference
IDENTITY
\[ 1 + \tan^2 x = \sec^2 x \]
THE TRICK
\[ \int \sec x \, dx = \ln | \sec x + \tan x | + C \]
Pro Tip:
If you are stuck with \(\int \tan x \, dx\), convert to \(\int \frac{\sin x}{\cos x} \, dx\) and use \( u = \cos x \).
Navigator Integral Worksheet Navigator's Integral
Integrating Powers of Tangent & Secant
DOC_REF: CALC_T2_L2
DATE: 2026.01.17
ID: ________________
u = tan x du = sec² x dx
u = sec x du = sec x tan x dx
Identity 1 + tan² x = sec² x
The Result ∫ sec x dx = ln|sec x + tan x|
Instructions: For each integral below, determine which "derivative pair" to use. Show your algebraic transformation before applying u-substitution.
1
The Standard Map Shift: \[ \int \tan^4(x) \sec^4(x) \, dx \]
Expected: u = tan x substitution
2
The Navigational Route: \[ \int \tan^3(x) \sec^5(x) \, dx \]
Expected: u = sec x substitution
3
Precision Calculation: \[ \int_{0}^{\pi/4} \tan^3(x) \, dx \]
Hint: tan³x = tan x (sec²x - 1)
The Mercator Verification
A navigator needs to find the total meridional part for a specific latitude range. Evaluate the following integral by first multiplying the numerator and denominator by \((\sec x + \tan x)\): \[ \int \sec(x) \, dx \] Demonstrate how this "multiplication by 1" leads to a simple logarithmic form.
Substitution Foundations Slides The Triangle Key
Introduction to Trigonometric Substitution
The Geometry of Algebra
How do we find the area of a circle using integration?
\[ A = 4 \int_{0}^{r} \sqrt{r^2 - x^2} \, dx \]
U-substitution fails here. But notice the structure: Pythagorean theorem!
r
x = r sin θ
The 3 Substitution Blueprints
Radical Form
\[ \sqrt{a^2 - x^2} \]
Substitute:
x = a sin θ
Radical Form
\[ \sqrt{a^2 + x^2} \]
Substitute:
x = a tan θ
Radical Form
\[ \sqrt{x^2 - a^2} \]
Substitute:
x = a sec θ
Step 1: Construct the Triangle
The Logic:
1 Identify the hypotenuse and legs based on the Pythagorean identity.
2 Label the angle \( \theta \).
3 Express the radical expression in terms of trig functions.
Leg 1 Leg 2 Hypotenuse θ
"The triangle is your bridge back to X."
From Nightmare to Dream
Algebraic Form
\[ \int \frac{dx}{x^2\sqrt{9-x^2}} \]
Trigonometric Form
\[ \int \frac{3\cos\theta \, d\theta}{(3\sin\theta)^2 \cdot 3\cos\theta} \]
The result? A simple \( \frac{1}{9} \int \csc^2 \theta \, d\theta \)
Substitution Blueprint Reference Sheet Substitution Blueprint
Core Reference Sheet
Master Key: TrigSub-01
Expression Substitution Differential (dx) Simplification \[ \sqrt{a^2 - x^2} \] \( x = a \sin \theta \) \( a \cos \theta \, d\theta \) \( a \cos \theta \) \[ \sqrt{a^2 + x^2} \] \( x = a \tan \theta \) \( a \sec^2 \theta \, d\theta \) \( a \sec \theta \) \[ \sqrt{x^2 - a^2} \] \( x = a \sec \theta \) \( a \sec \theta \tan \theta \, d\theta \) \( a \tan \theta \)
For \( x = a \sin \theta \)
\[ \sqrt{a^2 - x^2} \] x a θ
Leg: Adj = Radical, Opp = x
For \( x = a \tan \theta \)
a x \[ \sqrt{a^2 + x^2} \] θ
Leg: Adj = a, Opp = x
For \( x = a \sec \theta \)
a \[ \sqrt{x^2 - a^2} \] x θ
Leg: Adj = a, Opp = Radical
Blueprint Challenge
Identify the correct substitution and draw/label the reference triangle for the following integral. Do not solve yet.
\[ \int \frac{\sqrt{x^2 - 16}}{x^4} \, dx \]
Substitution
Identity Used
Draw & Label Triangle Here
Transform the Radical:
Transform \(dx\):
Translation Walkthrough Slides Triangle Translations
Solving the Full Trigonometric Substitution
Lesson 4
The 5-Step Translation Protocol
01
IDENTIFY
Choose the form based on the radical structure.
02
SUBSTITUTE
Replace \(x\), \(dx\), and the radical with trig terms.
03
INTEGRATE
Solve the resulting trigonometric integral.
04
TRIANGLE
Map the trig result back to sides of a triangle.
05
DECODE
Translate the trig answer back into algebra.
Crucial Step: Limits of Integration
When solving definite integrals, you have two choices:
Option A: Change Limits
Convert the \(x\) bounds into \(\theta\) bounds immediately using your substitution formula.
Option B: Translate Back
Ignore limits during integration, translate back to \(x\), then use original bounds.
Most students prefer Option A to avoid the messy triangle back-sub!
\[ x = 2\sin\theta \implies \theta = \arcsin(x/2) \]
Decoding back to X
If your trig integral result is:
\[ \frac{1}{25}\cot\theta + C \]
And your substitution was \(x=5\sin\theta\)...
\[ \sqrt{25-x^2} \] x 5 θ
Read the triangle: \(\cot\theta = \frac{\text{adj}}{\text{opp}}\)
\[ \frac{\sqrt{25-x^2}}{x} \]
Translation Challenge
"A secret message is encrypted in Trigonometry. Only the Triangle can decode it back to the reality of Algebra."
Encrypted Result:
\[ \frac{1}{2}\theta + \frac{1}{4}\sin(2\theta) + C \]
Hint: Use double-angle identities before translating back.
Translation Challenge Worksheet Translation Challenge
Trig to Algebra Decoding
UNIT_04_TRANS
Candidate ID
Mission Briefing
You are tasked with translating complex algebraic integrals into the language of trigonometry, solving them, and then decoding them back into their original algebraic form using reference triangles. Precision is vital; a single sign error can collapse the translation.
01
The Sine Shift: \[ \int \frac{x^2}{\sqrt{9-x^2}} \, dx \]
Target: Substitute \( x = 3\sin\theta \). Provide full back-substitution using the triangle.
Trig Integration Work
Reference Triangle
Back-Substitution & Final Result
02
The Tangent Trail: \[ \int_{0}^{1} \frac{dx}{(x^2+1)^2} \]
Target: Definite Integral. You may change limits or back-substitute.
Integration Workflow
03
The Secant Cipher: \[ \int \frac{\sqrt{x^2-25}}{x} \, dx \]
Target: Substitute \( x = 5\sec\theta \).
Primary Integration
Triangle
Final Decoded Answer
Advanced Radical Slides Mastery Lesson
Advanced Radicals
Completing the Square for Hidden Structures
Real-World Messiness
In structural engineering, finding the center of mass of a parabolic plate doesn't always result in a clean \( \sqrt{a^2 - x^2} \).
Instead, you might encounter something like:
\[ \int \frac{dx}{\sqrt{5 + 4x - x^2}} \]
This doesn't match our 3 blueprints... yet.
Structural Analysis
Requires Pre-Processing
The Square-Completion Protocol
1
Isolate: Group the \(x\)-terms and factor out any coefficient of \(x^2\).
2
Complete: Add and subtract \( (b/2)^2 \) inside the group to create a perfect square trinomial.
3
Reveal: Rewrite as \( (x \pm h)^2 \pm k \) to match a TrigSub form.
Example Transformation:
\[ 5 + 4x - x^2 \]
\[ = 5 - (x^2 - 4x) \]
\[ = 5 - (x^2 - 4x + 4) + 4 \]
\[ = 9 - (x - 2)^2 \]
Now it fits \( a^2 - u^2 \) with \( u = x - 2 \)!
The Shifted Triangle
When the radical is shifted (e.g., contains \(x-2\) instead of just \(x\)), the triangle rules stay the same!
If \( \sqrt{9 - (x-2)^2} \):
Let \( x - 2 = 3\sin\theta \)
Your legs will now be labeled with expressions like \(x-2\) and the radical.
\[ \sqrt{9-(x-2)^2} \] x - 2 3 θ
Mastery Checklist
Preparation
Is it a quadratic radical?
Does it need completing the square?
Factor out any \(a\) in \(ax^2\).
Execution
Let \( u = x \pm h \).
Don't forget \( dx = du \).
Back-substitute carefully.
Parabolic Plate Lab Worksheet Lab Report // Physics Application
Parabolic Plate Lab
STUDENT_ID __________________
SESSION_DATE 2026.01.17
Case Study: Center of Mass
An engineering firm is designing a thin parabolic metal plate. To determine the rotational stability, they must calculate the area-weighted integral of the plate's shape. The resulting integral for the plate's "Moment of Inertia" around a shifted axis is given by the expression below.
1
Phase I: Pre-Processing
Complete the square for the quadratic expression under the radical to reveal its trigonometric structure.
\[ \int \sqrt{3 - 2x - x^2} \, dx \]
Algebraic Transformation
Show Step-by-Step Completing the Square
Substitution Identity
Let u = ________
Trig Sub: u = ________
Reference Triangle
Label Triangle
2
Phase II: Execution & Final Report
Solve the transformed integral and back-substitute to find the general solution for the plate's area-weighted property.
Full Solution Workflow
Final Engineered Expression:
Mastery Check Slides Intro Mastery Check
Unit Assessment: Trig Integration & Substitution
The Assessment Blueprint
To demonstrate mastery of trigonometric integration, you will be evaluated on three distinct skill sets:
30%
Power Manipulation (Sine/Cosine/Secant/Tangent)
40%
Translation Accuracy (TrigSub & Triangles)
30%
Advanced Processing (Completing the Square)
Success Tip:
"The triangle is your most valuable asset. Label it before you integrate!"
Skill Check: Form Identification
\[ \sqrt{25 - x^2} \]
Sub Choice x = 5 sin θ
\[ \sqrt{x^2 - 16} \]
Sub Choice x = 4 sec θ
\[ \sqrt{x^2 + 9} \]
Sub Choice x = 3 tan θ
Matching the form is half the battle.
Expectations for Excellence
Criterion Standard for Mastery Work Organization Clearly labeled \(u\), \(du\), and trig-sub identities for every problem. Geometric Verification A correctly labeled reference triangle accompanies every TrigSub solution. Algebraic Integrity Accurate handling of constants and fractional coefficients during back-sub.
Go Forth
You have the blueprints. You have the triangles. Translate the world into calculus and back again.
Mastery Awaits
Trig Integration Mastery Assessment Mastery Assessment
Trig Integration & Substitution
STUDENT NAME ________________
DATE ________________
Testing Protocol: Show all work for full credit. Include clear u-substitution declarations and reference triangles where appropriate. Provide exact answers (in terms of \(\pi\), radicals, and fractions) unless otherwise specified.
Section I
Trigonometric Powers (20 Points)
1. Evaluate the integral: \[ \int \sin^2(x) \cos^3(x) \, dx \]
2. Solve: \[ \int \sec^4(x) \, dx \]
Section II
Trigonometric Substitution (50 Points)
3. Use Trigonometric Substitution to evaluate: \[ \int \frac{dx}{x^2\sqrt{x^2+4}} \]
Substitution & Integration Work
Triangle Label sides in terms of \(x\) and constants.
Final Decoded Answer (Variable \(x\))
4. Evaluate the definite integral: \[ \int_{0}^{2} \sqrt{4-x^2} \, dx \]
Section III
Advanced Processing (30 Points)
5. Complete the square and identify the correct Trigonometric Substitution for the integral below. (Do not fully integrate): \[ \int \frac{dx}{\sqrt{x^2 + 8x + 25}} \]
Completion of the Square
Chosen Substitution Identity
6. Challenge: Set up the integral for the surface area generated by rotating \( y = x^2 \) about the y-axis from \( y=0 \) to \( y=4 \). Simplify the integral until it is in a form ready for Trigonometric Substitution.
