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13. Integration
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Standard Level

13.06 Integrals resulting in logarithms

Lesson
Worksheet
Practice
Worksheet
Integration of 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{-2}{x}
b
\dfrac{6}{x}
c
\dfrac{1}{7 x}
d
\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{2}{x + 3} \, dx
f
\int \dfrac{2}{2 x + 3} \, dx
g
\int \dfrac{- 2}{x - 4} \, dx
h
\int \dfrac{1}{2 x + 3} \, dx
i
\int \dfrac{1}{5 - x} \, dx
j
\int \dfrac{2}{3 x + 5} \, dx
k
\int \dfrac{2 x}{x^{2} - 5} \, dx
l
\int \left(\dfrac{8}{x} + 5\right) \, dx
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_{3}^{5} \dfrac{x^{2} + 1}{x} \, dx
g
\int_{0}^{3} \dfrac{1}{5 + 4 x} \, dx
h
\int_{2}^{6} \left( 2 x + \dfrac{1}{x + 2}\right) \, dx
5

Determine the integral of the following functions:

a

f \left( x \right) = \dfrac{2 x^{2} + x - 3}{x}, for x \gt 0.

b

f \left( x \right) = \dfrac{5 x^{3} + 2 x^{2}}{x^{3}}, for x > 0.

6

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

a

Differentiate f \left( x \right) = x \ln x.

b

Hence, find \int \ln x \,dx.

Reverse chain rule
7

Consider the function f \left( x \right) = \ln \left(x^{2} - 4\right).

a

Use the chain rule to differentiate f \left( x \right).

b

Hence, find \int \dfrac{6 x}{x^{2} - 4} \, dx.

8

Consider the function f \left( x \right) = \ln \left(2x^{2} + 4x\right).

a

Use the chain rule to differentiate f \left( x \right).

b

Hence, find \int \dfrac{x+1}{2x^{2} + 4x} \, dx.

9

Consider the function f \left( x \right) = \ln \left(3x^{3} +5x^2\right).

a

Use the chain rule to differentiate f \left( x \right).

b

Hence, find \int \dfrac{18x^{2}+20x}{3x^{3}+5x^{2}} \, dx.

10

Find the integral of the following functions:

a

\int \dfrac{4}{4x - 3} \, dx

b

\int \dfrac{15}{3x - 2} \, dx

c

\int \dfrac{-8}{2x + 1} \, dx

d

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

e

\int \dfrac{3 x^2}{x^{3} + 1} \, dx

f

\int \dfrac{8 x}{2x^{2} - 3} \, dx

g

\int \dfrac{2 x^2}{2x^{3} + 7} \, dx

h

\int \dfrac{6 x^2+3}{2x^{3} + 3x} \, dx

Area under the curve
11

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

a

Sketch a graph of the function for x>0.

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 for x>0.

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 y>0.

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 y>0.

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 x>0.

b

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

17

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

a

State the domain of the function.

b

State the value the function approaches as x \to \infty.

c

Calculate the exact area enclosed by the function, the x-axis, and the lines x = 2 and x = 12.

18

Find the exact area enclosed by the function f \left( x \right) = x + \dfrac{5}{x + 5}, the x-axis and the lines x = 0 and x = 2.

19

Consider the function f \left( x \right) = \dfrac{6 x^{5} - 4 x^{2} + x}{2 x^{2}}, for x > 0.

a

Simplify and hence determine \int f \left( x \right) \, dx.

b

Find the exact area bounded by f \left( x \right), the x-axis and the lines x = 1 and x = 2.

20

Consider the functions y = x and y = \dfrac{3}{x}, for x \gt 0.

a

Sketch the graphs of these functions on the same coordinate axes.

b

At what x-value do the two functions intersect?

c

Find the area between the two functions, the x-axis and the line x = 3.

21

Consider the functions y = - x^{2} and y = \dfrac{1}{4 - x} for 0 \leq x \leq 3.

a

Sketch the graphs of these functions on the same coordinate axes.

b

Calculate the exact area bound by the curves between x = 1 and x = 3.

22

Consider the functions f \left( x \right) = \dfrac{1}{x} and g \left( x \right) = \sqrt{x}.

a

Sketch the graphs of the two functions on the same coordinate axes.

b

At what x-value do the two functions intersect?

c

Find the area of the region bounded by the two curves, the y-axis, and the line y = 2.

23

Consider the graph of the functions y = x^{2} and y = \dfrac{1}{x}:

Solve for the exact value of k such that: \int_{1}^{k} \dfrac{1}{x} \, dx = \int_{0}^{1} x^2 \, dx

-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
24

Consider the function f \left( x \right) = \ln \left(x^{2} + 1\right).

a

Differentiate f \left( x \right).

b

Hence, find \int \dfrac{x}{x^{2} + 1}\, dx.

c

Find the area bounded by g \left( x \right) = \dfrac{x}{ \left(x^{2} + 1\right)}, the x-axis and the lines x = 2 and x = 3.

25

Consider the function f \left( x \right) = \ln \left(\sin \left(x\right) + 1\right).

a

Differentiate f \left( x \right).

b

Hence, find \int \dfrac{\cos x}{\sin x + 1} \, dx.

c

Find the area bounded by g \left( x \right) = \dfrac{\cos x}{\sin \left(x\right) + 1}, the x-axis and the lines x = 0 and x = \dfrac{\pi}{6}.

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