Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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3:22 minutes

Problem 8.3.75c

Textbook Question

75. Exploring powers of sine and cosine
c. Prove that ∫₀ᵖⁱ sin²(nx) dx has the same value for all positive integers n.

Verified step by step guidance

Step 1: Begin by recalling the trigonometric identity for sine squared:

sin²(x) = (1 - cos(2x))/2

. This identity will simplify the integral.

Step 2: Substitute

sin²(nx)

in the integral with the identity:

∫₀ᵖⁱ sin²(nx) dx = ∫₀ᵖⁱ (1 - cos(2nx))/2 dx

. Split the integral into two parts:

∫₀ᵖⁱ (1/2) dx - ∫₀ᵖⁱ (cos(2nx)/2) dx

Step 3: Evaluate the first integral

∫₀ᵖⁱ (1/2) dx

. Since this is a constant, the result is straightforward:

(1/2) * x

evaluated from 0 to

π

Step 4: For the second integral

∫₀ᵖⁱ (cos(2nx)/2) dx

, recall that the integral of cosine over a full period (or multiples of

π

) is zero. Specifically,

∫₀ᵖⁱ cos(2nx) dx = 0

for any positive integer

n

Step 5: Combine the results from Step 3 and Step 4. The first integral contributes a constant value, while the second integral contributes zero. Therefore, the value of

∫₀ᵖⁱ sin²(nx) dx

is independent of

n

and remains the same for all positive integers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral calculates the accumulation of a function's values over a specific interval, providing the net area under the curve. In this case, the integral ∫₀ᵖⁱ sin²(nx) dx evaluates the area under the sine squared function from 0 to π, which is crucial for proving the equality for different values of n.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. The identity sin²(x) = (1 - cos(2x))/2 is particularly useful in this context, as it simplifies the integral of sin²(nx) and allows for easier evaluation across different integer values of n.

Periodicity of Sine Function

The sine function is periodic, meaning it repeats its values in regular intervals. For sin²(nx), the period is π/n, which implies that as n changes, the function still retains a consistent average value over the interval [0, π]. This periodicity is key to demonstrating that the integral yields the same result for all positive integers n.