New Zealand
Level 7 - NCEA Level 2

# Antidifferentiation

Lesson

We have already seen how to find the antiderivative of some very simple polynomials

For example, the antiderivative of a constant term $a$a, is $ax+C$ax+C

We can also see some examples of powers of $x$x

• Antiderivative of $x$x is $\frac{x^2}{2}+C$x22+C
• Antiderivative of $x^2$x2 is $\frac{x^3}{3}+C$x33+C
• Antiderivative of $x^3$x3 is $\frac{x^4}{4}+C$x44+C

Which leads to a general form of

• Antiderivative of $x^n$xn is $\frac{x^{n+1}}{n+1}+C$xn+1n+1+C

## Properties

• The antiderivative of a sum is the sum of the antiderivatives.

## $\int af(x)\pm bg(x)dx=a\int f(x)dx\pm b\int g(x)dx$∫af(x)±bg(x)dx=a∫f(x)dx±b∫g(x)dx

• Integral of a constant  $\int adx=ax+C$adx=ax+C
• All functions that are continuous at all points in an interval have antiderivatives that are defined on the interval.

## Using the properties

To find indefinite integrals of polynomial functions we combine the idea that the antiderivative of a sum is the same as the sum of the anitderivatives and our rule for finding the antiderivative of a power term together.

#### Example

Find 

 $\int3x^4+4x^3-2x^2+x-7dx$∫3x4+4x3−2x2+x−7dx $=$= $\int3x^4dx+\int4x^3dx-\int2x^2dx+\int xdx-\int7dx$∫3x4dx+∫4x3dx−∫2x2dx+∫xdx−∫7dx $=$= $3\int x^4dx+4\int x^3dx-2\int x^2dx+\int xdx-\int7dx$3∫x4dx+4∫x3dx−2∫x2dx+∫xdx−∫7dx $=$= $3\times\frac{x^5}{5}+4\times\frac{x^4}{4}-2\times\frac{x^3}{3}+\frac{x^2}{2}-7x+C$3×x55​+4×x44​−2×x33​+x22​−7x+C $=$= $\frac{3x^5}{5}+\frac{4x^4}{4}-\frac{2x^3}{3}+\frac{x^2}{2}-7x+C$3x55​+4x44​−2x33​+x22​−7x+C $=$= $\frac{3x^5}{5}+x^4-\frac{2x^3}{3}+\frac{x^2}{2}-7x$3x55​+x4−2x33​+x22​−7x

##### Example

A particular curve has a derivative given by $\frac{dy}{dx}=2x^4-3x^2+x$dydx=2x43x2+x

We are told that the curve passes through the point $(1,-\frac{1}{2})$(1,12).  Find the equation of the curve.

We start here with the indefinite integral

 $\int2x^4-3x^2+xdx$∫2x4−3x2+xdx $=$= $2\int x^4dx-3\int x^2dx+\int xdx$2∫x4dx−3∫x2dx+∫xdx $=$= $2\cdot\frac{x^5}{5}-3\cdot\frac{x^3}{3}+\frac{x^2}{2}+C$2·x55​−3·x33​+x22​+C $=$= $\frac{2x^5}{5}-x^3+\frac{x^2}{2}+C$2x55​−x3+x22​+C

We are given a set of conditions that can help us identify the value of $C$C.  We use the point $(1,-\frac{1}{2})$(1,12)

 $y$y $=$= $\frac{2x^5}{5}-x^3+\frac{x^2}{2}+C$2x55​−x3+x22​+C $-\frac{1}{2}$−12​ $=$= $\frac{2}{5}-1+\frac{1}{2}+C$25​−1+12​+C $-\frac{1}{2}-\frac{2}{5}+1-\frac{1}{2}$−12​−25​+1−12​ $=$= $C$C $C$C $=$= $-\frac{2}{5}$−25​

So the curve is $y=\frac{2x^5}{5}-x^3+\frac{x^2}{2}-\frac{2}{5}$y=2x55x3+x2225

#### Worked Examples

##### Question 1

Find the primitive function of $10x+7$10x+7.

Use $C$C as the constant of integration.

##### Question 2

Find the primitive function of $15x^4+16x^3$15x4+16x3.

Use $C$C as the constant of integration.

##### Question 3

Find an equation for $y$y if $\frac{dy}{dx}=\left(5x-2\right)\left(3x-4\right)$dydx=(5x2)(3x4).

Use $C$C as the constant of integration.

### Outcomes

#### M7-10

Apply differentiation and anti-differentiation techniques to polynomials

#### 91262

Apply calculus methods in solving problems