JZfa20 Math 1920 Name: ___________________

Trigonometric Substitution

Many important integrals involve a sum or difference of squares.

EXAMPLE: To compute the area A of the quarter unit disk (see figure),
we need to find the exact value of the integral:
1

A= 
0
12  x 2 dx

which involves the (square root of) a difference of squares.

Such a quantity can be represented geometrically

as a side of a right triangle:

Designating the base angle as 2, we can express

the quantities involving x in terms of 2 :
12  x 2

12  x 2
cos    12  x 2  cos 
1
x
sin    x  sin  ,
1
so dx  d sin  cos  d

Substitution of these quantities transforms the integral to 2 form:

1   2   2


0
2 2
1  x dx 

 cos   cos d     cos
0 0
2
 d 
   2
1 2 1 1  

20
1  cos 2 d     sin 2 
2  2 

4
 0

The above technique of trigonometric substitution is applicable to a wide variety of integrals that
involve a sum or difference of squares.
1
EXERCISE 1: Use trigonometric substitution to show that  2
dx  sin1 x  C
1 x
 Trigonometric substitution also works in absence of the square root:

1
EXAMPLE: To compute an expression for F ( x)   1 x 2 dx by trigonometric substitution,
i. Sketch a right triangle
and assign the appropriate quantities to its sides :
12  x 2

ii. Use the triangle to express the integrand as a function of 2:

2
1  1 
2     cos 2 
1 x  1 x2 
iii. Use your triangle to express x as a function of 2 :
x
tan   x  tan
1

iv. Write the differential dx as an expression in 2 :

dx  d tan  sec 2  d

v. Convert the integral to 2form and compute the result:

1
F ( x)   dx   cos
 sec
2 2

  d   d    C
1 x2 1

vi. Use your triangle to convert the resulting expression to x:

x  tan    tan1 x

vii. The conclusion confirms the well-know fact that

1
F ( x)   1 x 2 dx    C  tan1 x  C

EXERCISE 2. Use trigonometric substitution to that :

1 1 x a
a 2
x 2 dx  ln
2a x  a
C
EXERCISES:
3. Use trigonometric substitution to show that
1
 x x 2  1 dx  sec 1 x  C

4. Use trigonometric substitution to show that

2
1 1
x 2 2
dx 
4
 5 2
1 4 x
5. Use trigonometric substitution to show that
1
x2  3
0 4 x 2
dx 
3

2
6. Using trigonometric substitution calculate the area bounded by the ellipse
x2 y 2
E: 2  2  1
a b
in the first quadrant:

a. Show that the area can be expressed as:

a
b
a 0
A a2  x 2 dx

b. Calculate the area of the ellipse

7. a. Use trigonometric substitution to show that:


1
1 x 2  
dx  ln x  1  x 2  C

1
 1 x2
dx 

d sinh1 x 1
b. Show that  .
dx 1 x2

1
Conclusion:  1 x2
dx 

c. From (a) and (b), we may conclude that

sinh1 x 
8. Use trigonometric substitution to show that
3
x2 9
0 9 x 2
dx 
4


9. Use trigonometric substitution to show that

2
1 3
x
1
2
4 x 2
dx 
4
10. Use trigonometric substitution to show that
2
4  x2 
1
x2
dx  3
3
11. Use trigonometric substitution to show that
2

 4  x 2 dx  2 2  2ln 2  1
0
12. Use trigonometric substitution to show that
a
a2  x 2 
a
 x2
dx  3
3
2
EXERCISE 2 - Solution: Use trigonometric substitution to that :

1 1 x a
a 2
x 2 dx  ln
2a x  a
C

Solution:

Referring to the triangle:

2 2
1  1  1 a  1
2      2 sec 
2
2  2 
a x  a x 
2 2 a  a x 
2 2 a
and
x
 sin  dx  a cos  d
a

we express the integral in trigonometric form as:

1 1 1
a 2
x 2 dx 
a 2 ( a)  sec  cos  d 
2
a
sec  d 

1 1 a x
ln sec   tan  C  ln 2  C
a a a  x2 a2  x 2
1 a x 1 x a 1 x a
ln  C  ln C ln C
a ( a  x)( a  x) a x a 2a x  a