What is the formula for antiderivatives?
What is the formula for antiderivatives?
An antiderivative of a function f(x) is a function whose derivative is equal to f(x). That is, if F′(x)=f(x), then F(x) is an antiderivative of f(x). x33,x33+1,x33−42,x33+π. x33+c,where c is a constant….Exercise 6.
| Function | General antiderivative | Comment |
|---|---|---|
| xn | 1n+1xn+1+c | for n,c any real constants with n≠−1 |
What is an antiderivative in simple terms?
An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.
Can a function have two Antiderivatives?
Thus any two antiderivative of the same function on any interval, can differ only by a constant. The antiderivative is therefore not unique, but is “unique up to a constant”. The square root of 4 is not unique; but it is unique up to a sign: we can write it as 2.
What are the Antiderivatives of trig functions?
This Section: 4. Integrals of Trigonometric Functions
| Derivative Rule | Antiderivative Rule |
|---|---|
| d dx sin x = cos x | cos x dx = sin x + C |
| d dx cos x = − sin x | sin x dx = − cos x + C |
| d dx tan x = sec2x | sec2x dx = tan x + C |
| d dx cotan x = − cosec2x | cosec2x dx = − cotan x + C |
Why do we differentiate and integrate?
Differentiation is used to calculate instant velocity. It is also used to find whether a function is increasing or decreasing. Integration is used to calculate the area of curved surfaces. It is also used to calculate the volume of objects.
What are the rules of antiderivative?
To find antiderivatives of basic functions, the following rules can be used:
- xndx = xn+1 + c as long as n does not equal -1. This is essentially the power rule for derivatives in reverse.
- cf (x)dx = c f (x)dx.
- (f (x) + g(x))dx = f (x)dx + g(x)dx.
- sin(x)dx = – cos(x) + c.
What are derivatives and Antiderivatives?
A derivative refers to finding the rate of change of a function. It can be defined as . The operator used is most often . An antiderivative is the opposite of a derivative. It’s also known as the integral, and often refers (graphically) to the area under a graph.
What is the most general antiderivative of F?
If F is an antiderivative of f, we say that F(x) + C is the most general antiderivative of f and write The symbol ∫ is called an integral sign, and ∫f(x)dx is called the indefinite integral of f. is the most general antiderivative of f. If F is an antiderivative of f, then
What are the common uses of antiderivatives?
Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations. A differential equation is an equation that relates an unknown function and one or more of its derivatives. The equation
Are there any other antiderivatives of Yes?
Consider the function Knowing the power rule of differentiation, we conclude that is an antiderivative of since Are there any other antiderivatives of Yes; since the derivative of any constant is zero, is also an antiderivative of Therefore, and are also antiderivatives.
What is the family of antiderivatives of 2x?
For example, since x2 is an antiderivative of 2x and any antiderivative of 2x is of the form x2 + C, we write The collection of all functions of the form x2 + C, where C is any real number, is known as the family of antiderivatives of 2x. Figure 4.85 shows a graph of this family of antiderivatives.
