INTEGRATION 02

Antiderivatives

An antiderivative reverses differentiation: from a known derivative, we recover a function that could have produced it.

The main idea

Suppose

\[ F'(x)=f(x). \]

Then \(F(x)\) is called an antiderivative of \(f(x)\).

\[ \boxed{ F'(x)=f(x) \quad\Longleftrightarrow\quad F(x)\text{ is an antiderivative of }f(x) } \]

1. Reversing differentiation

We know that

\[ \frac{d}{dx}x^3=3x^2. \]

Therefore, if we start with

\[ 3x^2, \]

one antiderivative is

\[ x^3. \]

So differentiation and antidifferentiation move in opposite directions:

\[ x^3 \quad\xrightarrow{\ d/dx\ }\quad 3x^2 \] \[ 3x^2 \quad\xrightarrow{\ \text{antidifferentiate}\ }\quad x^3. \]

2. But the antiderivative is not unique

Consider

\[ x^3, \qquad x^3+2, \qquad x^3-7. \]

Differentiating each gives

\[ 3x^2. \]

This happens because the derivative of every constant is zero.

Therefore \(3x^2\) has infinitely many antiderivatives.

3. The constant of integration

We represent all possible antiderivatives by writing

\[ \boxed{ \int 3x^2\,dx=x^3+C } \]

where \(C\) is an arbitrary constant.

The symbol \(C\) is called the constant of integration.

4. Why \(C\) is necessary

Suppose

\[ F(x)=x^3+C. \]

Then

\[ F'(x)=3x^2 \]

for every constant \(C\).

Differentiation loses information about additive constants. Antidifferentiation therefore cannot recover which constant was originally present.

5. Indefinite integral notation

The notation

\[ \int f(x)\,dx \]

means the family of antiderivatives of \(f(x)\).

Thus,

\[ \boxed{ \int f(x)\,dx=F(x)+C } \]

whenever

\[ F'(x)=f(x). \]

This is called an indefinite integral.

6. The reverse power rule

The differentiation rule

\[ \frac{d}{dx}x^{n+1} = (n+1)x^n \]

can be reversed.

Therefore,

\[ \boxed{ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C, \qquad n\neq-1 } \]

To integrate a power of \(x\):

increase the power by \(1\), then divide by the new power.

7. Example

Consider

\[ \int x^4\,dx. \]

Increase the power:

\[ 4+1=5. \]

Then divide by \(5\):

\[ \boxed{ \int x^4\,dx = \frac{x^5}{5}+C } \]

8. Check by differentiating

We can check the answer:

\[ \frac{d}{dx} \left( \frac{x^5}{5}+C \right) = \frac{1}{5}(5x^4)+0. \]

Therefore,

\[ \frac{d}{dx} \left( \frac{x^5}{5}+C \right) = x^4. \]

So the antiderivative is correct.

9. Constant multiples

Constants can be kept outside the integration process.

For example,

\[ \int 6x^2\,dx = 6\int x^2\,dx. \]

Therefore,

\[ 6\left(\frac{x^3}{3}\right)+C = 2x^3+C. \]

\[ \boxed{ \int 6x^2\,dx = 2x^3+C } \]

10. Integrating term by term

Sums and differences can be integrated term by term.

Consider

\[ \int(3x^2+4x-5)\,dx. \]

Integrate each term:

\[ \int3x^2\,dx + \int4x\,dx - \int5\,dx. \]

This gives

\[ x^3+2x^2-5x+C. \]

Therefore,

\[ \boxed{ \int(3x^2+4x-5)\,dx = x^3+2x^2-5x+C } \]

11. Integrating a constant

If \(c\) is constant, then

\[ \boxed{ \int c\,dx=cx+C } \]

because

\[ \frac{d}{dx}(cx)=c. \]

12. Negative powers

The reverse power rule also works for negative powers, except \(x^{-1}\).

For example,

\[ \int x^{-2}\,dx. \]

Increase the power:

\[ -2+1=-1. \]

Then divide by \(-1\):

\[ \boxed{ \int x^{-2}\,dx = -x^{-1}+C = -\frac{1}{x}+C } \]

13. Fractional powers

Roots can be written as powers.

For example,

\[ \sqrt{x}=x^{1/2}. \]

Therefore,

\[ \int\sqrt{x}\,dx = \int x^{1/2}\,dx. \]

Increase the power:

\[ \frac12+1=\frac32. \]

Then divide by \(3/2\):

\[ \boxed{ \int\sqrt{x}\,dx = \frac{2}{3}x^{3/2}+C } \]

14. Why \(n=-1\) is different

The power rule formula

\[ \frac{x^{n+1}}{n+1} \]

cannot be used when

\[ n=-1, \]

because then

\[ n+1=0, \]

and division by zero is undefined.

Instead,

\[ \boxed{ \int\frac{1}{x}\,dx = \ln|x|+C } \]

for \(x\neq0\).

15. Some basic antiderivatives

\[ \boxed{ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C, \qquad n\neq-1 } \] \[ \boxed{ \int\frac{1}{x}\,dx = \ln|x|+C } \] \[ \boxed{ \int e^x\,dx = e^x+C } \] \[ \boxed{ \int\cos x\,dx = \sin x+C } \] \[ \boxed{ \int\sin x\,dx = -\cos x+C } \]

16. Derivative and antiderivative

Differentiation asks:

\[ \boxed{ \text{How is this function changing?} } \]

Antidifferentiation asks the reverse question:

\[ \boxed{ \text{What function has this rate of change?} } \]

For example,

\[ F'(x)=2x \]

leads to

\[ F(x)=x^2+C. \]

17. Connection with accumulation

At first, integration appears to have two different meanings:

one as accumulation, and another as antidifferentiation.

These ideas are not separate.

The Fundamental Theorem of Calculus shows that accumulated change can be calculated using antiderivatives.

\[ \boxed{ \text{accumulation} \longleftrightarrow \text{antiderivative} } \]