IXL - Integrate by completing the square (Calculus practice)

Find the indefinite integral. Use

to represent an arbitrary constant.

∫

–

ref_doc_title.

Learn with an example

question

Find the indefinite integral. Use

to represent an arbitrary constant.

∫

key idea

To complete the square for an expression of the form

add

to make

Remember to also subtract

to compensate for adding that term.

–

Here are some formulas for integrals that involve inverse trigonometric functions.

∫
d
x
1
–
x
2
=
sin
-1
x
+
C
∫
d
x
x
2
+
1
=
tan
-1
x
+
C
∫
d
x
x
x
2
–
1
=
sec
-1
absolute value bar
x
absolute value bar
+
C

solution

You were asked to find the following indefinite integral.

∫

This integral does not resemble any familiar integration formulas, and there is no obvious substitution. However, there is a quadratic expression in the denominator, so try rewriting it by completing the square.

Add and subtract

(

)

–

Add and subtract

(

)

–

(

)

Rewrite the integral.

∫

(

)

The rewritten integral looks similar to

∫

This suggests using the substitution

∫

(

)

∫

tan

-1

Rewrite using

tan

-1

(

)

Rewrite using

$Confetti$

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Teacher tools

Work it out

Not feeling ready yet? These can help:

O.4Complete the square

U.2Integrals involving inverse trigonometric functions

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