topic badge

4.06 The definite integral

Worksheet
Definite integrals
1

Evaluate the following definite integrals:

a

\int_{0}^{3} \left( 4 x + 5\right) dx

b

\int_{ - 2 }^{0} \left( 10 x + 4\right) dx

c

\int_{2}^{4} \left( 6 x + 5\right) dx

d

\int_{3}^{4} \left( - 4 x + 3\right) dx

e

\int_{ - 4 }^{5} \left( - 8 x + 3\right) dx

f

\int_{ - 10 }^{ - 6 } \left( - 5 x + 8\right) dx

g

\int_{ - 2 }^{0} 9 x^{2} dx

h

\int_{ - 1 }^{3} 9 x^{2} dx

i

\int_{ - 2 }^{1} \left(x^{2} + 4\right) dx

j

\int_{ - 1 }^{2} \left( 9 x^{2} + 1\right) dx

k

\int_{4}^{6} \left( 9 x^{2} + 2 x + 7\right) dx

l

\int_{ - 4 }^{2} x \left(x - 4\right) dx

m

\int_{0}^{4} 5 x^{\frac{3}{2}} dx

n

\int_{ - 3 }^{6} x \left(x + 3\right) \left(x - 6\right) dx

o

\int_{6}^{11} \sqrt{x - 2} dx

p

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

q

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

r

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

s

\int_{1}^{2} \dfrac{x^{5} - x^{ - 2 }}{x^{2}} dx

Properties of definite integrals
2

Consider the function f \left( x \right) = x^2. Find the value of the following:

a
\int_{1}^{1} f \left( x \right) dx.
b
\int_{5}^{5} f \left( x \right) dx.
c
\int_{-2}^{-2} f \left( x \right) dx.
d
\int_{a}^{a} f \left( x \right) dx.
3

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

a

Find the value of \int_{0}^{3} f \left( x \right) dx.

b

Find the value of \int_{0}^{3} 5f \left( x \right) dx.

c

Find the value of 5\int_{0}^{3} f \left( x \right)dx.

d

State the property of definite integrals demonstrated by parts (b) and (c).

4

Given that \int_{ - 2 }^{7} f \left( x \right) dx = 2, find \int_{ - 2 }^{7} 5 f \left( x \right) dx.

5

Given that \int_{ - 1 }^{6} f \left( x \right) dx = 3, find \int_{ - 1 }^{6} \left( 9 f \left( x \right) - 2\right) dx.

6

Consider the function f \left( x \right) = 6 x.

a

Find the value of \int_{4}^{8} f \left( x \right) dx.

b

Find the value of \int_{8}^{4} f \left( x \right) dx.

c

State the property of definite integrals demonstrated by parts (a) and (b).

7

Given that \int_{4}^{6} f \left( x \right) dx = 3, find the values of the following:

a

\int_{6}^{4} f \left( x \right) dx

b

\int_{4}^{6} 3 f \left( x \right) dx

c

\int_{4}^{6} \left(f \left( x \right) + x\right) dx

8

Given that \int_{ - 1 }^{2} f \left( x \right) dx = 4 and \int_{2}^{8} f \left( x \right) dx = 8, find the values of the folowing:

a

\int_{ - 1 }^{8} f \left( x \right) dx

b

\int_{8}^{ - 1 } f \left( x \right) dx

c

\int_{ - 1 }^{2} 2 f \left( x \right) dx + \int_{2}^{8} 3 f \left( x \right) dx

9

Given that \int_{ - 1 }^{3} f \left( x \right) dx = 5 and \int_{2}^{3} f \left( x \right) dx = 2, find the values of the following:

a

\int_{ - 1 }^{2} f \left( x \right) dx

b

\int_{3}^{ - 1 } f \left( x \right) dx

c

2 \int_{ - 1 }^{2} f \left( x \right) dx + \int_{2}^{3} 3 f \left( x \right) dx

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

ACMMM128

recognise and use the additivity and linearity of definite integrals

ACMMM131

understand the formula ∫ {from a to b} f(x)dx= F(b)−F(a) and use it to calculate definite integrals

What is Mathspace

About Mathspace