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10.06 Anti-differentiation

Worksheet
Anti-differentiation
1

For each of the following derivatives, find an equation for the primitive function. Use c as the constant of integration:

a
\dfrac{d y}{d x} = 8 x
b
\dfrac{d y}{d x} = 9
c
\dfrac{d y}{d x} = x \sqrt{x}
d
\dfrac{d y}{d x} = x^{2} \left( 10 x^{2} - 9 x\right)
e
\dfrac{d y}{d x} = \dfrac{x^{3} + 4}{x^{3}}
f
\dfrac{d y}{d x} = \sqrt{x}
g
\dfrac{dy}{dx} = \left( 5 x - 2\right) \left( 3 x - 4\right)
h
\dfrac{d y}{d x} = 18 \sqrt{x}
i
\dfrac{dy}{dx} = \left(x + 4\right) \left(x + 6\right)
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} = \dfrac{6}{\sqrt{x}}
m
\dfrac{d y}{d x} = x^{ - \frac{3}{7} } + x^{ - \frac{2}{5} }
n
\dfrac{d y}{d x} = \dfrac{x^{6}}{4} + \dfrac{x^{2}}{3}
o
\dfrac{d y}{d x} = 4 x^{\frac{2}{5}} + 3 x^{\frac{4}{7}}
p
\dfrac{d y}{d x} = 8 x^{3} + 3 x^{\frac{5}{3}} - 3
2

Find the following indefinite integrals:

a

\int 4 \ dx

b

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

c

\int x^{5} dx

d

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

e

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

f

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

g

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

h

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

i

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

j

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

k

\int x^{ - 5 } dx

l

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

m

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

n

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

o

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

p

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

q

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

r

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

s

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

3

Find the equation of a curve p given that \dfrac{d p}{d t} = 6 t - 5, and when t = 3, \dfrac{d p}{d t} = 13 and p = 15.

4

Find the equations of the curve y given that:

a

\dfrac{d y}{d x} = 4 x + 7 and the curve passes through the point \left(3, 41\right).

b

\dfrac{d y}{d x} = 9 x^{2} - 10 x + 2 and the curve passes through the point \left(2, 13\right).

c

\dfrac{d y}{d x} = 10 x^{4} + 20 x^{3} + 6 x^{2} + 6 x + 9 and the curve passes through the point \left( - 3 , - 133 \right).

d

\dfrac{d y}{d x} = 9 x^{\frac{2}{3}} and the curve passes through the point \left(8, \dfrac{889}{5}\right).

5

Find the equation of the curve that has a gradient of 15 x^{2} + 7 and passes through the point \left(2, 59\right).

Kinematics
6

The velocity v \left( t \right), in \text{m/s}, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t, \text{ where } t \geq 0The object is initially at the origin.

a

Find the displacement x \left( t \right)= \int v(t) \, dt of the particle at time t.

b

Find the time at which x(t) = 54 \text{ m}.

7

The velocity v, in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 6 t + 10, \text{ where } t \geq 0

a

Find the displacement s \left( t \right) of the particle at time t, given that the object starts its movement at 8 \text{ m} to the right of the origin.

b

Find the displacement of the object after 5 seconds.

8

The velocity v \left( t \right), in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} + 30 t + 9, \text{ where } t \geq 0The object starts its movement at 6 \text{ m} to the left of the origin.

a

Find the displacement, s \left( t \right), of the particle at time t.

b

Find the displacement of the object after 5 seconds.

9

The velocity of a particle moving in rectilinear motion is given by v \left( t \right) = 6 t^{2} - 14 t + 3, where v \left( t \right) is the velocity in metres per second and t is the time in seconds.

The displacement after 2 seconds is 4 \text{ m} to the left of the origin.

a

Calculate the initial velocity of the particle.

b

Find the function x \left( t \right) for the position of the particle. Use C as the constant of integration.

c

Calculate the displacement of the particle after 4 seconds.

d

Assuming the object continues to move in the same direction, find the total distance travelled between 2 and 4 seconds.

10

The velocity of a particle moving in rectilinear motion is given by v \left( t \right) = 4 t - 4 where v is the velocity in metres per second from the origin and t is the time in seconds. The particle is instantaneously stationary when it is 1 \text{ m} right of the origin.

a

Find the time when the particle is stationary.

b

Find the function x \left( t \right) for the position of the particle.

c

Find the value of x(t) at:

i

t=0

ii

t=1

iii

t=2

11

The velocity v \left( t \right) of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} - 48 t, \text{ where } t \geq 0The object starts its movement 5 \text{ m} to the right of the origin.

a

Find the displacement, x \left( t \right), of the particle at time t.

b

Find the times, t, when the object is at rest.

c

Find the displacement at which the object is stationary other than its initial position.

12

The velocity v \left( t \right), in \text{ m/s}, of an object along a straight line after t seconds is modelled by v \left( t \right) = 12 \sqrt{t}The object is initially 5 \text{ m} to the right of the origin.

a

Find the function x \left( t \right) for the position of the particle.

b

Hence, calculate the position of the object after 9 seconds.

13

The acceleration a in \text{m/s}^{2} of an object travelling horizontally along a straight line after t seconds is modelled by a = 2 t - 15 where t \geq 0. The object is initially moving to the right at 56 \text{ m/s}.

a

Find the velocity v of the particle at time t.

b

Find the times, t, at which the particle is at rest.

14

The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a \left( t \right) = 6 t - 27, \text{ where } t \geq 0After 10 seconds, the object is moving at 90 \text{ m/s} in the positive direction.

a

State the velocity, v, of the particle at time t.

b

Find all the times at which the particle is at rest.

15

The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a = 2, \text{ where } t \geq 0The object is initially 8 \text{ m} to the right of the origin and moving to the left at 5 \text{ m/s}.

a

Find the velocity, v, of the particle at time t.

b

Find the displacement, x, of the particle at time t.

c

Find the position of the object at 7 seconds.

d

Find the time at which the particle is moving at a speed of 3 \text{ m/s} to the right.

Applications
16

An ice cube with a side length of 25 \text{ cm} is removed from the freezer and starts to melt at a rate of 25 \text{ cm}^{3}/\text{min}. Let V be its volume t minutes after it is removed from the freezer.

a

State the equation for the rate of change of volume.

b

State the equation for the volume, V, of the cube as a function of time t.

c

Find the value of t when the ice cube has melted completely.

17

In a closed habitat, the population of kangaroos P \left( t \right) is known to increase according to the function P' \left( t \right) = \dfrac{t}{2} + 9 where t is measured in months since counting began.

a

Calculate the total change in the population of kangaroos in the first 4 months since counting began.

b

Find the number of months it will take from when counting began for the population of kangaroos to increase by 88.

18

The total revenue, R (in thousands of dollars), from producing and selling a new product, t weeks after its launch, is given by \dfrac{d R}{d t} = 401 + \dfrac{500}{\left(t + 1\right)^{3}}

a

Given that the initial revenue at the time of launch was zero, state the revenue function.

b

Find the average revenue earned over the first 5 weeks.

c

Calculate the revenue earned in the 6th week.

19

Wheat is poured from a silo into a truck at a rate of \dfrac{d M}{d t} , where \dfrac{d M}{d t} = 81 t - t^{3} \text{ kg/s} and t is the time in seconds after the wheat begins to flow.

a

Find an expression for the mass M \text{ kg} of wheat in the truck after t seconds, if initially there was 1 tonne of wheat in the truck.

b

Calculate the total mass of wheat in the truck after 8 seconds.

c

Find the largest value of t for which the expression for \dfrac{d M}{d t} is physically possible.

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calculate anti-derivatives of polynomial functions

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