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VCE 12 Methods 2023

7.01 Introduction to anti-differentiation

Worksheet
Reverse of differentiation
1

Consider the function x^{4}.

a

Calculate \dfrac{d}{dx} \left(x^{4}\right).

b

Hence find \int 4 x^{3} dx.

2

Consider the function x^{6}.

a

Calculate \dfrac{d}{dx} \left(x^{6}\right).

b

Hence find \int 24 x^{5} dx.

3

Consider the function x^{5} + x^{4}.

a

Calculate \dfrac{d}{dx} \left(x^{5} + x^{4}\right).

b

Find \int \left( 35 x^{4} + 28 x^{3}\right) dx.

4

Consider the function x^{ - 4 }.

a

Calculate \dfrac{d}{dx} \left(x^{ - 4 }\right).

b

Hence find \int 16 x^{ - 5 } dx.

5

Consider the function x^{8} + x^{6}.

a

Calculate \dfrac{d}{dx} \left(x^{8} + x^{6}\right).

b

Find \int \left( 4 x^{7} + 3 x^{5}\right) dx.

6

Consider the function \sqrt[5]{x^{6}}.

a

Calculate \dfrac{d}{dx} \left(\sqrt[5]{x^{6}}\right).

b

Hence find \int 6 \sqrt[5]{x} dx.

Anti-differentiation
7

Consider the gradient function f \rq \left( x \right) = 2.

a

Find the general antiderivative to f \rq \left( x \right).

b

The gradient function f \rq \left( x \right), is a constant function. What type of function is the antiderivative function, f \left( x \right).

c

If the form of a constant gradient function is f \rq \left( x \right) = m , write the general form of the antiderivative function.

8

Consider the gradient function f \rq \left( x \right) = - 8 x.

a

Find the general antiderivative to f \rq \left( x \right).

b

The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).

c

If the form of a linear gradient function is f \rq \left( x \right) = ax , write the general form of the antiderivative function.

9

Consider the gradient function f \rq \left( x \right) = 4 x + 3.

a

Find the general antiderivative to f \rq \left( x \right).

b

The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).

c

If the form of a linear gradient function is f \rq \left( x \right) = ax +b , write the general form of the antiderivative function.

10

Consider the gradient function f \rq \left( x \right) = 7 x^{2}.

a

Find the general antiderivative to f \rq \left( x \right).

b

The gradient function f \rq \left( x \right), is a quadratic function. What type of function is the antiderivative function, f \left( x \right).

c

If the form of a quadratic gradient function is f \rq \left( x \right) = ax^2 , write the general form of the antiderivative function.

11

Find the general antiderivative of the following:

a

\dfrac{d y}{d x} = 9

b

\dfrac{d y}{d x} = 8 x

c

\dfrac{d y}{d x} = 10 x + 7

d

\dfrac{d y}{d x} = 9 x^{2}

e

\dfrac{d y}{d x} = 9 x^{2} + 4 x - 6

f

\dfrac{d y}{d x} = x^{5}

g

\dfrac{d y}{d x} = 15 x^{4} + 16 x^{3}

h

\dfrac{d y}{d x} = \dfrac{x^{6}}{4} + \dfrac{x^{2}}{3}

i

\dfrac{d y}{d x} = x^{ - 6 }

j

\dfrac{d y}{d x} = \dfrac{15}{x^{6}}

k

\dfrac{d y}{d x} = \dfrac{10}{x^{6}} - \dfrac{9}{x^{4}}

l

\dfrac{d y}{d x} = 4 x^{\frac{2}{5}} + 3 x^{\frac{4}{7}}

m

\dfrac{d y}{d x} = x^{ - \frac{3}{7} } + x^{-\frac{2}{5} }

n

\dfrac{d y}{d x} = 8 x^{3} + 3 x^{\frac{5}{3}} - 3

o

\dfrac{d y}{d x} = \dfrac{x^{3} + 4}{x^{3}}

p

\dfrac{d y}{d x} = x^{2} \left( 10 x^{2} - 9 x\right)

q

\dfrac{dy}{dx} = \left( 5 x - 2\right) \left( 3 x - 4\right)

r

\dfrac{dy}{dx} = \left(x + 4\right) \left(x + 6\right)

s

\dfrac{d y}{d x} = \sqrt{x}

t

\dfrac{d y}{d x} = 18 \sqrt{x}

u

\dfrac{d y}{d x} = x \sqrt{x}

v

\dfrac{d y}{d x} = \dfrac{6}{\sqrt{x}}

Indefinite integrals
12

Find the following indefinite integrals:

a

\int 4 \ dx

b

\int x^{3} dx

c

\int 4 x^{3} dx

d

\int \dfrac{1}{4} x^{2} dx

e

\int x^{5} dx

f

\int \left(x^{2} + 4 x\right) dx

g

\int \left(5 - 3 t - 4 t^{2}\right) dt

h

\int \left( 4 x^{3} + 3 x^{2}\right) dx

i

\int \left( 3 x^{2} + 6 x + 3\right) dx

j

\int \left( 3 x + 3 x^{2} + x^{3}\right) dx

k

\int \left(\dfrac{x^{2}}{5} + \dfrac{x^{3}}{4} + 3\right) dx

l

\int x^{ - 5 } dx

m

\int \left( 3 x^{ - 2 } - 8 x\right) dx

n

\int \dfrac{5}{x^{2}} dx

o

\int x^{\frac{1}{4}} dx

p

\int x^{\frac{4}{5}} dx

q

\int 5 x^{\frac{3}{4}} dx

r

\int \sqrt{x} \ dx

s

\int \left(x^{\frac{4}{5}} + 5 x^{\frac{2}{3}}\right) dx

t

\int \left( 5 x^{\frac{7}{3}} + \dfrac{9}{15} x^{\frac{5}{4}}\right) dx

u

\int \left( 5 x^{\frac{2}{3}} + 3 \sqrt{x} + 6\right) dx

v

\int \dfrac{1}{\sqrt{x}} \ dx

w

\int \left( 3 x - 2\right) \left(x + 6\right) dx

x

\int x \left( 4 x + 7\right) \left( 7 x + 2\right) dx

y

\int x^{2} \left( 7 x^{4} + 9\right) dx

z

\int 7 x \left(x - 1\right)^{2} dx

13

Simplify and hence find the following indefinite integrals:

a

\int x \left( 7 x^{3} + 3\right)^{2} dx

b

\int \left(\sqrt{x} - 2\right)^{2} dx

c

\int \left(\sqrt{x} + \dfrac{5}{x}\right)^{2} dx

d

\int \sqrt{x} \left( 7 x^{2} + 5 x\right) dx

e

\int \left(\dfrac{x^{3}}{3} - \dfrac{3}{x^{3}}\right) dx

f

\int \dfrac{5 x^{5} - 2 x}{x} dx

g

\int \dfrac{8 x^{6} + 9 x}{x^{5}} dx

h

\int \dfrac{x^{3} + 5}{\sqrt{x}} dx

i

\int y \sqrt{y} \ dy

j

\int \dfrac{1}{y \sqrt{y}} \ dy

k

\int a y^{3} dy \ , a is a constant

l

\int \dfrac{1}{n} \sqrt{t} \ dt \ , n is a non-zero constant

Applications
14

For each of the following gradient functions:

i

State what type of function the antiderivative, f\left(x\right), is.

ii

Find the general antiderivative to f \rq \left( x \right).

iii

Sketch a possible graph for the antiderivative f\left(x\right).

a
f\rq\left(x\right)=6
b
f'\left(x\right)=8 x + 8
c
f'\left(x\right)=7 x^{2}
d
f'\left(x\right)=- 5 x \left(x + 2\right)
15

Consider the gradient function f'\left(x\right)=\dfrac{3}{x^{2}}.

a

Find the general antiderivative to f \rq \left( x \right).

b

Sketch a possible graph of the antiderivative f\left(x\right).

16

The gradient function, f'\left(x\right), has only one x-intercept at \left( - 4 , 0\right), a y-intercept at \left(0, - 3 \right) and a constant gradient.

a

Find f'\left(x\right).

b

Find the general antiderivative to f \rq \left( x \right).

c

Sketch a possible graph of the antiderivative f\left(x\right).

Equation of the antiderivative
17

Consider the equation \dfrac{d y}{d x} = 4 x + 7.

a

Find a general equation for y.

b

Find the equation of y, if the curve passes through the point \left(3, 41\right).

18

Consider the equation \dfrac{d y}{d x} = 9 x^{2} - 10 x + 2.

a

Find a general equation for y.

b

Find the equation of y, if the curve passes through the point \left(2, 13\right).

19

A family of curves has a gradient function y \rq = 15 x^{2} + 7.

a

Find the equation of y for the family of curves.

b

Find the equation of the curve that passes through the point \left(2, 59\right).

20

Consider the gradient function \dfrac{d y}{d x} = 10 x^{4} + 20 x^{3} + 6 x^{2} + 6 x + 9.

a

Find an equation for y.

b

Find the equation of y, if the curve passes through the point \left( - 3 , - 133 \right).

21

Consider the gradient function \dfrac{d y}{d x} = 9 x^{\frac{2}{3}}.

a

Find an equation for y.

b

Find the equation of y, if the curve passes through the point \left(8,\dfrac{889}{5}\right).

22

Find the equation of a curve p in terms of t, given the following:

  • \dfrac{d p}{d t} = 6 t - 5
  • when t = 3, \dfrac{d p}{d t} = 13 and p = 15

23

Find the equation of a curve y in terms of x, given the following:

  • y \rq \rq = 0.
  • y contains the points \left(-1, -1 \right) and \left(0, 2 \right).
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Outcomes

U34.AoS3.6

anti-derivatives of polynomial functions and functions of the form f(ax+b) where f is x^n, e^x, sin(x), cos(x) and linear combinations of these

U34.AoS3.19

find anti-derivatives of polynomial functions and power functions, functions of the form f(ax+b) where f is x^n, e^x, sine or cosine, and simple linear combinations of these, using pattern recognition, or by hand

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