# Primitive Function 3(sin(3t) + cos(3t))

• Attis
In summary, the conversation discusses finding the primitive function for v(t)=3(sin(3t)+cos(3t)) with the initial condition s(0)=0. The correct primitive function is s(t)=-cos(3t)+sin(3t)+C, and the mistake made by the speaker was confusing the derivative with the anti-derivative.
Attis

## Homework Statement

Find the primitive function s(t) for v(t)=3(sin(3t) + cos(3t)) for which s(0)=0

## The Attempt at a Solution

v(t) =3(sin(3t) + cos(3t)) = 3sin3t +3cos3t

s(t) =cos3t -sin3t + C

0 = cos3(0) - sin3(0) + C
0= 1 + C
C=-1
such that my primitive function is cos3t - sin3t - 1

s(t) = -cos(3t) + sin(3t) + C
and s(0) = 0 = -1 + 0 + C such that C= 1 and the primitive function is sin3t -cos3t + 1

has he made a mistake? or am I confused?

You've made mistakes. What is the primitive function of cos and sin?

1 person
the primitive function is the anti-derivative. so cosx becomes sinx and sinx becomes -cosx?
s(t)=-cos3t+sin3t + C

Now I got it. I´d gotten confused between the derivative and the anti-derivative. Thanks for confirming that I´d made the mistake!

## What is a primitive function?

A primitive function, also known as an antiderivative, is the inverse operation of differentiation. It is a function that, when differentiated, returns the original function.

## What is the difference between a primitive function and an indefinite integral?

A primitive function is the most general form of an indefinite integral. It includes a constant of integration, while an indefinite integral does not.

## How do you find the primitive function of 3(sin(3t) + cos(3t))?

To find the primitive function, use the power rule to integrate each term separately. The primitive function of sin(3t) is -cos(3t), and the primitive function of cos(3t) is sin(3t). Therefore, the primitive function of 3(sin(3t) + cos(3t)) is -3cos(3t) + 3sin(3t) + C, where C is the constant of integration.

## Can the primitive function of 3(sin(3t) + cos(3t)) be simplified further?

No, the primitive function cannot be simplified any further. It is the most general form of the indefinite integral of 3(sin(3t) + cos(3t)).

## Why is it important to find the primitive function of a function?

Finding the primitive function allows us to find the area under a curve, which has many applications in science, engineering, and mathematics. It is also a fundamental concept in calculus and is used to solve differential equations.

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