Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

문제 8.3.75c

8장, 문제 8.3.75c

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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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