Download Section 5.7b Worksheet 3 (for the 4th edition of the book)

MA 182
Trigonometric Substitutions
th
Section 5.7 (4 Edition)
d (tan 1 x)
1
d (sin 1 x)
1
Recall that
and


2
dx
1  x2
dx
1 x
Using these basic formulas,
d (sin 1 x / a)
1/ a
1


2
2
dx
x
a  x2
1 2
a
and
d (tan 1 x / a )
1/ a
a

 2
2
x
dx
a  x2
1 2
a
Based on these differentiation formulas, we have the following integration formulas:

1
a2  x2
dx  sin 1
x
C
a
and
a
2
1
1
x
dx  tan 1  C
2
x
a
a
Examples:
(1)

(2)
x
1
9 x
2
2
dx  sin 1
x
C
3
1
1
x
dx 
tan 1
C
2
2
2
Other integrals which contains expressions of the form
a2  x2 or
a2  x2 (a > 0) can often
be evaluated by making use of a substitution which involves appropriate inverse trigonometric
functions. The basic idea is to make a substitution that will eliminate the radical.
If the integrand contains
a2  x2 , a substitution involving the inverse sine function will often
be useful. In this case,
x
x
Let   sin 1 , ( / 2     / 2) , so that sin   .
a
a
Therefore x  a sin  , dx  a cos  d and
a2  x2  a2  a2 sin 2   a2 (1  sin 2  )  a cos2   a cos  a cos
Note that cos  cos because  / 2     / 2 and cos  0 in quadrants I and IV.
The following example illustrates the technique.

x2
4  x2
dx
If the integrand contains
a2  x2 , a substitution involving the inverse tangent function will
often be useful.
In this case,
Let   tan 1
x
x
,  / 2     / 2 , so that tan   .
a
a
Therefore x  a tan  , dx  a sec2  , and
a2  x2  a2  a2 tan 2   a 1  tan 2   a sec2   a sec  a sec
Note that sec  sec because  / 2     / 2 and sec  0 in quadrants I and IV.
An example follows.
x
3
x 2  9 dx
MA 182
Trigonometric Substitutions - Exercises
Use either a trigonometric or another appropriate substitution to evaluate each indefinite integral.
1.

2.

3.

1
dx
16  x 2
x
9 x
4
x3
9  x4
dx Hint: Let u  x 2
dx Hint: This can be integrated using a trigonometric substitution but there
is a much easier way.
4.

5.
p. 394/#16
6.
x
2
7.
x
2
8.
(a)
Verify by differentiation that  sec d  ln sec  tan   C
(b)
Evaluate the indefinite integral
4  x 2 dx
1
dx
 25
x
dx
 25
9.
p. 394/#13
10.
p. 394/#14

1
x 4
2
dx
Hint: After using the trigonometric substitution x  2 tan  , rewrite the
integral in terms of sines and cosines in order to integrate.