   Chapter 3.9, Problem 16E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) f ( t ) = 3 cos t − 4 sin t

To determine

To find:

The most general antiderivative of the given function.

Explanation

1) Concept:

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is,Fx+C where C  is an arbitrary constant.

Definition:

A function F  is called an antiderivative of f on an interval I if

F'x=fx for all x in I.

2) Formula:

The particular antiderivative of cosx  is sinx

And the particular antiderivative of sinx  is -cosx

Power rule of antiderivative

ddx xn+1n+1=xn

3) Given:

ft=3cost-4sint

4) Calculation:

Given function ft=3cost-4sint

The particular antiderivative of cosx  is sinx

And the particular antiderivative of sinx  is -cosx

Using these particular antiderivatives,

The antiderivative of the function

ft=3cost-4sint  can be written as

3 (sint)-4 ( -cost)+C

- 4·(-cost)  becomes 4cost

Therefore, the antiderivative can be written as

3sint+4cost+<

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