MAC2312-3
Integration Using
Trigonometric Substitution
PURPOSE
Sometimes it is not possible to use a u substitution in an integration problem. When an
integrand contains x2 + a2 but there is no way to obtain an x for replacing x dx by du, we
may be able to use the trig identity:
1 + tan2x = sec2x or more generally a2 + (a tan x)2 = (a sec x)2
When an integrand contains x2 − a2, we may be able to use the trig identity:
sin2x + cos2x = 1 that is a2 − (a sin x)2 = (a cos x)2
PROCEDURE
In this section, we see how to integrate expressions like
We substitute the following to simplify the expressions to be integrated:
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 1 Example 1:
with a = 3
We simplify the denominator of the question before proceeding to integrate:
Now, substituting
and
into the given integral gives us:
(‘K’ is a constant)
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 2 We now need to get our answer in terms of x (since the question was in terms of x).
Since we let x = 3 tan θ, we get
We can now draw a triangle to find the value of sin θ :
Hence, we notice that
Therefore, we can conclude that:
Example 2
Let x = 4 sec θ then dx = 4 sec θ tan θ dθ and x2 = 16 sec2θ
Simplifying the square root part:
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 3 Substituting dx = 4 sec θ tan θ dθ, x2 = 16 sec2x and √
integral gives us:
16
4 tan θ into the given
Since we let x = 4 sec θ, we get
Using a triangle, we can also derive that:
and
Therefore, we can conclude that:
NOTE: We could have changed upper
and lower limits for θ here, and there
would be no need to convert our
expression back in terms of x.
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 4 Example 3
Let x = 2 sin θ, so dx = 2 cos θ dθ
Example 4
Firstly, note that
If we put u = x + 1, then du = dx and our integral becomes:
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 5 Now, we use u = sec θ and so: du = sec θ tan θ dθ
Returning to our integral, we have:
REVIEW
The exact substitution used depends on the form of the integral:
(a2−x2)n
(x2+a2)n
(x2−a2)n
x=a sin
x= a tan
x=a sec
dx=a cos
dx=a sec2
dx=a sec tan
Document created by South Campus Library Learning Commons 011/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed. 6