The antiderivative

Calculus, grade 11 (4 units) Lesson 16 of 20Learning

The antiderivative

The inverse of differentiating

An antiderivative of $\displaystyle f(x)$ is a function $\displaystyle F(x)$ whose derivative is $\displaystyle f(x)$ : $\displaystyle {F'(x)}={f(x)}$ .
For example, for $\displaystyle {f(x)}={2x}$ :

$\displaystyle {F(x)}={x^2}$ is an antiderivative, since $\displaystyle {(x^2)'}={2x}$ ;
$\displaystyle x^2+5$ and $\displaystyle x^2-3$ are antiderivatives too: the derivative of a constant is 0.

All the antiderivatives of $\displaystyle f$ are $\displaystyle F(x)+C$ , $\displaystyle C$ any number – the constant of integration. Their graphs are the same graph, shifted up or down.

The indefinite integral

All the antiderivatives of $\displaystyle f$ are written as an indefinite integral:

\displaystyle \int f(x)\,dx=F(x)+C

$\displaystyle \int$ – the integral sign;
$\displaystyle dx$ – the variable is $\displaystyle x$ ;
for example, $\displaystyle {\int2x\,dx}={x^2+C}$ .

The check: differentiate the answer, and the function under the integral sign must come out.

The rules of integration

Each rule is a rule of differentiation, the other way round:

\displaystyle \int x^{n}\,dx

\displaystyle =

\displaystyle \frac{x^{n+1}}{n+1}+C

\displaystyle n\neq-1

a number multiplying the function stays: $\displaystyle {\int5x^2\,dx}={\frac{5x^3}{3}+C}$ ;
the integral of a sum is the sum of the integrals;
$\displaystyle {\int k\,dx}={kx+C}$ .

And from the derivative $\displaystyle {\left(-\frac{c}{a(ax+b)}\right)'}={\frac{c}{(ax+b)^2}}$ :

\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx

\displaystyle =

\displaystyle -\frac{c}{a(ax+b)}+C

For example, $\displaystyle {\int\frac{6}{(2x+1)^2}\,dx}={-\frac{6}{2(2x+1)}+C}={-\frac{3}{2x+1}+C}$ .

Example

In the example below we work out $\displaystyle \int(3x^2-4x+1)\,dx$ term by term:

\int(3x^2-4x+1)\,dx=\frac{3x^3}{3}-\frac{4x^2}{2}+x+C=x^3-2x^2+x+C

The check: $\displaystyle {(x^3-2x^2+x+C)'}={3x^2-4x+1}$ .

An antiderivative through a point

The derivative of $\displaystyle f$ is $\displaystyle {f'(x)}={2x-4}$ , and the graph of $\displaystyle f$ passes through the point $\displaystyle A(3,5)$ .
In the solution below we find $\displaystyle f$ from its derivative, and $\displaystyle C$ from the point:

$\displaystyle {f(x)}={\int(2x-4)\,dx}={x^2-4x+C}$ ;
substitute $\displaystyle A$ : $\displaystyle {3^2-4\cdot3+C}={5}$ , that is $\displaystyle {-3+C}={5}$ , so $\displaystyle {C}={8}$ ;
$\displaystyle {f(x)}={x^2-4x+8}$ .

\displaystyle y=x^{2}-4x

\displaystyle y=x^{2}-4x+4

\displaystyle f(x)=x^{2}-4x+8

\displaystyle A=(3,5)

The board shows three of the antiderivatives of $\displaystyle 2x-4$ . All of them have the same slope at every $\displaystyle x$ , and only one passes through $\displaystyle A$ .

Exercise 1

Work out the integrals.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Multiply out the brackets, and then work out the integral.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Work out the integrals with the rule for $\displaystyle \frac{c}{(ax+b)^2}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

$\displaystyle {F'(x)}={3x^2+2}$ , and the graph of $\displaystyle F$ passes through the point $\displaystyle A(1,4)$ .

$\displaystyle {F(x)}={x^3+2x+C}$ . Work out $\displaystyle C$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write $\displaystyle F(x)$ on the board.

\displaystyle A=(1,4)

Exercise 5

$\displaystyle {f'(x)}={\frac{2}{(x-1)^2}}$ , and $\displaystyle {f(2)}={1}$ .
Find $\displaystyle f(x)$ , and write it on the board. Work out $\displaystyle C$ in the card under the board.

\displaystyle B=(2,1)

\displaystyle x=1

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$