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Chapter 17: Problem 8

Find \(\int 3 \cos n \pi x \mathrm{~d} x\).

Short Answer

Expert verified
Question: Find the integral of the function \(f(x) = 3 \cos n \pi x\). Answer: The integral of the given function is \(\int 3 \cos n \pi x \mathrm{~d} x = \frac{3}{n \pi} \sin n \pi x + C\).

Step by step solution

01

Identify the function to integrate

First, let's identify the function that needs to be integrated. In this problem, it is given as \(f(x) = 3 \cos n \pi x\).
02

Apply the integration rule for cosine functions

Recall that the integral of \(\cos(ax)\) with respect to \(x\) is \(\frac{1}{a} \sin(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration. In our case, the function to integrate is \(3 \cos n \pi x\), so we have \(a = n \pi\).
03

Integrate the function

Using the integration rule for cosine functions, we can integrate \(3 \cos n \pi x\) as follows: \(\int 3 \cos n \pi x \mathrm{~d} x = 3 \int \cos n \pi x \mathrm{~d} x = 3 \left(\frac{1}{n \pi} \sin n \pi x + C\right) = \frac{3}{n \pi} \sin n \pi x + C\).
04

Write the final answer

The final answer for the integral of the given function is: \(\int 3 \cos n \pi x \mathrm{~d} x = \frac{3}{n \pi} \sin n \pi x + C\).

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Most popular questions from this chapter

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$

Find \(\int 3 \mathrm{e}^{2 x} \mathrm{~d} x\).

Find \(\int \frac{1}{s^{2}-2 s+5} \mathrm{~d} s\).

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Find \(\int \sqrt{\frac{x}{x-2}} \mathrm{~d} x\).
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