Chapter 17: Problem 10

Find \(\int \frac{1}{2} \sin \frac{2 \pi t}{T} \mathrm{~d} t\).

Short Answer

Expert verified

Question: Determine the integral of the function \(f(t) = \frac{1}{2} \sin \frac{2 \pi t}{T}\). Answer: The integral of the given function is \(-\frac{T}{4\pi} \cos \frac{2\pi t}{T} + C\).

Step by step solution

Identify the function and its variable

The function is given as \(f(t) = \frac{1}{2} \sin \frac{2 \pi t}{T}\) where \(t\) is the variable and \(T\) is a constant parameter.

Rewrite using a substitution

We observe that there is a composition of functions: \(\sin\) of \(\frac{2\pi t}{T}\). Let's use substitution to simplify the integral. Let \(u = \frac{2\pi t}{T}\), and differentiate to find \(\mathrm{d}u = \frac{2\pi}{T} \mathrm{d}t\). We can rewrite \(\mathrm{d}t\) in terms of \(\mathrm{d}u\). So, \(\mathrm{d}t = \frac{T}{2\pi}\mathrm{d}u\). Now, the integral becomes: \(\int \frac{1}{2} \sin u \frac{T}{2\pi} \mathrm{d}u\).

Factor out constants

We can factor out the constant \(\frac{T}{4\pi}\) from the integral: \(\frac{T}{4\pi} \int \sin u \mathrm{d}u\).

Integrate the sine function

The antiderivative of \(\sin u\) is \(-\cos u\). So the integral becomes: \(-\frac{T}{4\pi} \cos u + C\).

Replace \(u\) with the original variable \(t\)

We used the substitution \(u = \frac{2\pi t}{T}\), so we need to express the answer in terms of \(t\). Hence, the solution to the integral is: \(-\frac{T}{4\pi} \cos \frac{2\pi t}{T} + C\).

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Find \(\int \frac{1}{2} \sin \frac{2 \pi t}{T} \mathrm{~d} t\).

Short Answer

Step by step solution

Identify the function and its variable

Rewrite using a substitution

Factor out constants

Integrate the sine function

Replace \(u\) with the original variable \(t\)

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