IXL - Properties of definite integrals (Calculus practice)

The function

is integrable on

[

]

with

∫

(

)

–

and

∫

(

)

–

Evaluate the definite integrals.

∫

(

)

∫

(

)

∫

(

)

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Learn with an example

question

The function

is integrable on

[

–

]

with

∫

(

)

–

and

∫

–

(

)

Evaluate the definite integrals.

∫

(

)

∫

–

(

)

∫

–

(

)

key idea

Suppose the functions

and

are integrable on an interval that contains constants

and

Let

be any constant. The following properties hold.

∫
a
a
f
(
x
)

d
x
=
0
∫
b
a
f
(
x
)

d
x
=
–
∫
a
b
f
(
x
)

d
x
∫
b
a
k
f
(
x
)

d
x
=
k
∫
b
a
f
(
x
)

d
x
∫
b
a
[
f
(
x
)
±
g
(
x
)
]

d
x
=
∫
b
a
f
(
x
)

d
x
±
∫
b
a
g
(
x
)

d
x
∫
b
a
f
(
x
)

d
x
=
∫
c
a
f
(
x
)

d
x
+
∫
b
c
f
(
x
)

d
x

solution

Use properties to evaluate the definite integrals.

First, evaluate

∫

(

)

Use the given information that

∫

(

)

–

∫

(

)

Apply Property 3

∫

(

)

Apply Property 3

Substitute

(

–

)

Substitute

–

Next, evaluate

∫

–

(

)

Use the given information that

∫

–

(

)

∫

–

(

)

Apply Property 2

–

∫

–

(

)

Apply Property 2

Apply Property 3

∫

–

(

)

Apply Property 3

Substitute

(

)

Substitute

Finally, evaluate

∫

–

(

)

∫

–

(

)

Apply Property 5

∫

–

(

)

∫

(

)

Apply Property 5

Substitute

(

–

)

Substitute

–

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