Topics
9. Further Integral Calculus
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AustraliaNSW
Year 12

9.02 Integration involving logarithmic functions

Lesson
Worksheet
Practice
Worksheet
Reciprocal functions
1

Write an equation for f(x), given that the derivative of f \left( x \right) is \dfrac{1}{x}.

2

Find the primitive function of the following:

a
\dfrac{6}{x}
b
\dfrac{1}{7 x}
c
\dfrac{1}{x + 5}
3

Find the following indefinite integrals:

a
\int \dfrac{4}{x} \, dx
b
\int \dfrac{- 7}{x} \, dx
c
\int \dfrac{3}{4 x} \, dx
d
\int \dfrac{1}{x - 4} \, dx
e
\int \dfrac{\, dx}{x + 2}
f
\int \dfrac{2}{x + 3} \, dx
g
\int \dfrac{2}{2 x + 3} \, dx
h
\int \dfrac{- 2}{x - 4} \, dx
i
\int \dfrac{1}{2 x + 3} \, dx
j
\int \dfrac{1}{5 - x} \, dx
k
\int \dfrac{2}{3 x + 5} \, dx
l
\int \dfrac{2 x}{x^{2} - 5} \, dx
m
\int \dfrac{3 x^{2}}{x^{3} - 5} \, dx
n
\int \dfrac{6 x^{2}}{x^{3} + 3} \, dx
o
\int \dfrac{e^{x}}{e^{x} + 2} \, dx
p
\int \dfrac{x}{x^{2} - 3} \, dx
q
\int \dfrac{4 x - 4}{x^{2} - 2 x} \, dx
r
\int \left(\dfrac{8}{x} + 5\right) \, dx
s
\int \dfrac{9 y}{y^{2} + 1} dy
4

Find the exact value of the following definite integrals:

a
\int_{3}^{9} \dfrac{3}{x} \, dx
b
\int_{4}^{6} \dfrac{1}{x - 2} \, dx
c
\int_{3}^{5} \dfrac{2}{x + 2} \, dx
d
\int_{\frac{8}{3}}^{\frac{17}{3}} \dfrac{4}{3 x - 2} \, dx
e
\int_{1}^{3} \dfrac{5}{x + 1} \, dx
f
\int_{0}^{3} \dfrac{16}{4 x + 3} \, dx
g
\int_{3}^{5} \dfrac{x^{2} + 1}{x} \, dx
h
\int_{7}^{10} \dfrac{2 x - 9}{x^{2} - 9 x + 23} \, dx
i
\int_{0}^{3} \dfrac{1}{5 + 4 x} \, dx
j
\int_{2}^{7} \dfrac{8 x + 24}{x^{2} + 6 x + 12} \, dx
k
\int_{2}^{6} \left( 2 x + \dfrac{1}{x + 2}\right) \, dx
l
\int_{2}^{4} \dfrac{x + 2}{x^{2} + 4 x + 8} \, dx
m
\int_{5}^{7} \left(\dfrac{1}{x - 3} - 2\right) \, dx
5

Consider the function y = x \ln x - x.

a

Find an expression for y'.

b

Hence find \int \ln x \text{ } dx.

c

Find the exact value of \int_{\sqrt{e}}^{e} \ln x \text{ } dx.

6

Consider the function y = x^{2} e^{x}.

a

Find an expression for \dfrac{dy}{dx}.

b

Hence find \int x \left( 2+ x \right) e^{x} \text{ } dx.

7

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

a

Find an expression for f'\left( x \right) .

b

Hence evaluate \int_{\sqrt{e}}^{e} \dfrac{ \ln x}{x} \text{ } dx.

8

Consider the function y = 2 x^{2} \ln x - x^{2}.

a

Show that y' = 4x \ln x.

b

Hence find the primitive function of x \ln x.

c
Find the exact value of \int_{e}^{2} x \ln x \text{ }dx.
9

Consider the function y = \ln \left( \ln x \right).

a

Find an expression for y'.

b

Hence find the primitive function of \dfrac{1}{x \ln x}.

10

Given that \int_{0}^{2} \dfrac{e^{x}}{e^{x} + 3} \, dx = \ln k, find the exact value of k.

Area under a curve
11

Consider the function y = \dfrac{2}{x}.

a

Sketch a graph of the function.

b

Find the exact area bounded by the curve, the x-axis, and the lines x = 2 and x = 3.

12

Consider the function y = \dfrac{1}{x - 2}.

a

Sketch a graph of the function.

b

Find the exact area bounded by the curve, the x-axis, and the lines x = 3 and x = 5.

13

Consider the function y = \dfrac{1}{x + 3}.

a

Sketch a graph of the function.

b

Find the exact area bounded by the curve, the x-axis, and the lines x = 2 and x = 4.

14

Consider the function y = \dfrac{1}{2 x + 5}.

a

Sketch a graph of the function.

b

Find the exact area enclosed by the curve, the x-axis, and the lines x = 3 and x = 4.

15

Consider the function y = \dfrac{1}{4 - 3 x}.

a

Sketch a graph of the function.

b

Find the exact area enclosed by the curve, the x-axis, and the lines x = 3 and x = 5.

16

Consider the function y = \dfrac{2}{3 x + 5}.

a

Sketch a graph of the function.

b

Find the exact area enclosed by the curve, the coordinate axes and the line x = 4.

17

Find the exact area enclosed by the function f \left( x \right) = \dfrac{x}{x^{2} + 2}, the x-axis and the lines x = 2 and x = 4.

18

Find the exact area under the curve y = \dfrac{x + 5}{x^{2} + 10 x + 1} between x = 0 and x = 6.

19

Consider the function y = \ln 2 x.

a

State the domain of the function.

b

Find the derivative function, y'.

c

For what values of x is y' positive?

d

Find the exact area bounded by the curve y = \ln 2 x, the x-axis and the line x = e.

Points on a curve
20

Find the equation of the curve f \left( x \right), given the derivative function and a point on the curve:

a

f' \left( x \right) = \dfrac{x}{x^{2} - 7}, and f \left( 4 \right) = \ln 3.

b

f' \left( x \right) = \dfrac{5}{5 x - 4}, and the point \left(3, \ln 11\right).

c

f' \left( x \right) = \dfrac{2}{x +1}, and the point \left(0, 1\right).

d

f'(x) = \dfrac{x^2+x+1}{x} and f \left( 1 \right)= 1\dfrac{1}{2}

21
a

Find f \left( x \right) given that f''\left( x \right) = \dfrac{1}{x^2} , f'\left( 1 \right) = 0 and f \left( 1 \right) = 3.

b

Hence find the exact value of f \left( e \right).

22

Consider y' = \dfrac{ 2x + 5 }{ x^{2} + 5x + 4}, where function y has a y-intercept of \, 1 + \ln 4.

Find the exact value of y when x = 1.

23

Consider the derivative function y' = \dfrac { 2 + x }{x}.

a

Find function y, given that it passes through \left(1, 1\right).

b

Find the exact value of y, when x = 2.

24

Find g \left( x \right) given that g' \left( x \right) = \dfrac{2x^{3} - 3x - 4}{x^{2}} and g \left( 2 \right) = -3 \ln 2 .

Applications
25

A particle moves so that its velocity, in metres per second, over time t is given by:

v = \dfrac{2t+3}{t^2+3t+1}

If the particle's initial displacement, x, is 5 metres, find x when t = 3 seconds. Round your answer to two decimal places.

26

A circus tent is 7 \text{ m} high and has a radius of 6 \text{ m}. The equation to describe the curved roof of the tent is y = \dfrac{7}{x + 1}, as shown in the diagram:

Calculate the exact cross-sectional area of the tent.

27

A group of rhinos are introduced into a reserve park and their population is observed over time. The rate of growth of the number of rhinos is found to be modelled by: \dfrac{dP}{dt} = \dfrac{4 t}{t^{2} + 2} where P is the population of rhinos in the reserve, t years after the program started.

a

Considering the graph of \dfrac{dP}{dt}, describe the rate of change of the population of rhinos.

b

Use the model to predict the total whole number of rhinos that were born in the reserve between 2 and 4 years after the program started.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
t
0.5
1
1.5
2
\dfrac{dP}{dt}
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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-7

applies the concepts and techniques of indefinite and definite integrals in the solution of problems

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