Antiderivatives and indefinite integrals

Reversing derivatives

If differentiation finds a rate of change, integration can recover the original quantity from its rate. An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$:

$F'(x)=f(x)$

For example, since

rac{d}{dx}x^2 = 2x

an antiderivative of $2x$ is $x^2$.

Indefinite integrals are written

$int f(x),dx = F(x) + C$

where $C$ is the constant of integration.

Why the constant matters

The derivative of a constant is zero. Therefore many different functions can have the same derivative. For example,

rac{d}{dx}(x^2)=2x

rac{d}{dx}(x^2+5)=2x

rac{d}{dx}(x^2-100)=2x

All have derivative $2x$. That is why the indefinite integral includes $+C$:

$int 2x,dx = x^2 + C$

In physics, $C$ often represents an initial condition. If velocity tells how position changes, integration can recover position only up to an unknown starting position.

Basic antiderivative rules

The power rule for integration reverses the derivative power rule:

int x^n,dx = rac{x^{n+1}}{n+1}+C

for $n eq -1$.

For example:

int x^2,dx = rac{x^3}{3}+C

$int 5x^4,dx = x^5 + C$

The special case $n=-1$ gives

int rac{1}{x},dx = ln|x|+C

Common antiderivatives

Some important antiderivatives include:

$int e^x,dx = e^x + C$

$int cos x,dx = sin x + C$

$int sin x,dx = -cos x + C$

Again, trig formulas assume angles are in radians.

Recovering position from velocity

If velocity is known as a function of time, position can be found by integrating:

$x(t)=int v(t),dt$

More completely,

$x(t)=int v(t),dt + C$

The constant $C$ is determined by initial position.

For example, if

$v(t)=3t^2$

then

$x(t)=t^3+C$

If $x(0)=5$, then $C=5$, so

$x(t)=t^3+5$

Recovering velocity from acceleration

Similarly, if acceleration is known, velocity comes from integrating acceleration:

$v(t)=int a(t),dt$

If $a(t)=g$ is constant, then

$v(t)=gt+C$

The constant is the initial velocity $v_0$, so

$v(t)=v_0+gt$

This is one of the standard constant-acceleration relationships.

Meaning of $dx$

In the integral symbol $int f(x),dx$, the $dx$ indicates the variable of integration. It tells you what variable is being accumulated over. In physics, this matters because integrating with respect to time is different from integrating with respect to position.

The big idea

Indefinite integration finds families of functions whose derivatives are known. The constant of integration represents missing starting information. In physics, antiderivatives let you recover position from velocity, velocity from acceleration, and accumulated quantities from rates.