Math 113 Weierstrass Substitution Techniques

Math 113 Weierstrass Substitution Techniques

The Weierstrass substitution is a powerful technique in calculus for integrating rational functions of trigonometric functions. It utilizes the substitution u = tan(x/2) to simplify complex integrals. This method provides substitutions for all six trigonometric functions, making it easier to solve integrals using partial fractions. The document includes detailed derivations and examples, illustrating how to apply the Weierstrass substitution effectively. Ideal for students in Math 113 or anyone studying integral calculus, this resource enhances understanding of trigonometric integration techniques.

Key Points

  • Explains the Weierstrass substitution for integrating trigonometric functions.
  • Derives substitutions for sin x, cos x, and other trigonometric identities.
  • Includes detailed examples demonstrating the application of the Weierstrass method.
  • Suitable for Math 113 students and those studying integral calculus.
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  • newtopiccyclegrowin
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Math 113
The Weierstrass Substitution
The Weierstrass substitution enables any rational function of the regular six trigonometric functions to
be integrated using the methods of partial fractions. It uses the substitution of
u = tan
x
2
. (1)
The full method are substitutions for the values of dx, sin x, cos x, tan x, csc x, sec x, and cot x. Using
the identity tan
2
θ + 1 = sec
2
θ, the derivative of (1) is
du =
1
2
sec
2
x
2
dx =
1
2
h
1 + tan
2
x
2
i
dx =
1
2
1 + u
2
dx.
It follows that
dx =
2 du
1 + u
2
. (2)
To derive the substitutions for sin x and the
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
u
√
1 + u
2
x
2
Figure 1: Reference triangle for u = tan(
x
2
)
other trigonometric substitutions, refer to figure
1 and use the double angle identitities for sin x
and cos x. The double angle identity for sin x is
sin x = 2 sin
x
2
cos
x
2
and for cos x, the double angle identity is
cos x = cos
2
x
2
− sin
2
x
2
.
The substitution for sin x is
sin x = 2 sin
x
2
cos
x
2
= 2
u
√
1 + u
2
1
√
1 + u
2
=
2u
1 + u
2
(3)
Similarly, for cos x, it is
cos x = cos
2
x
2
− sin
2
x
2
=
1
√
1 + u
2
2
−
u
√
1 + u
2
2
=
1 − u
2
1 + u
2
(4)
By using (3) and (4), the substitutions for tan x, csc x, sec x, and cot x is
tan x =
sin x
cos x
=
2u
1+u
2
1−u
2
1+u
2
=
2u
1 − u
2
csc x =
1
sin x
=
1 + u
2
2u
cot x =
1
tan x
=
1 − u
2
2u
sec x =
1
cos x
=
1 + u
2
1 − u
2
(5)
Note that the resulting equations for all 6 trigonometric functions, along with dx all are simple
polynomials in u. Hence, integrals of rational functions of trigonometric functions can be solved using
partial fractions. In summary,
u = tan
x
2
sin x =
2u
1 + u
2
csc x =
1 + u
2
2u
tan x =
2u
1 − u
2
dx =
2 du
1 + u
2
cos x =
1 − u
2
1 + u
2
sec x =
1 + u
2
1 − u
2
cot x =
1 − u
2
2u
The Weierstrass Substitution, page 2
Example:
A short example can illustrate the power of the method. The integral of sec x is known to be
Z
sec x dx = ln |sec x + tan x| + C,
which is found by employing the “trick” of mutiplying the integrand by
sec x+tan x
sec x+tan x
and employing the u
substitutiton u = sec x + tan x. A straightforward solution can be found using the Weierstrass method.
It follows that
Z
sec x dx =
Z
1 + u
2
1 − u
2
2
1 + u
2
dx
=
Z
2
1 − u
2
du
=
Z
1
1 − u
+
1
1 + u
du
= −ln |1 − u| + ln |1 + u| + C
= ln
1 + u
1 − u
+ C
= ln
1 + tan
x
2
1 − tan
x
2
+ C
Note that this does not look like ln |sec x + cos x|. However, using a bit of trickery (multiply by one)
always helps!
Z
sec x dx = ln
1 + u
1 − u
+ C
= ln
1 + u
1 − u
·
1 + u
1 + u
+ C
= ln
(1 + u)
2
1 − u
2
+ C
= ln
1 + 2u + u
2
1 − u
2
+ C
= ln
1 + u
2
1 − u
2
+
2u
1 − u
2
+ C
= ln |sec x + tan x| + C
Hence, the two solutions are identical!
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End of Document
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FAQs of Math 113 Weierstrass Substitution Techniques

What is the Weierstrass substitution used for?
The Weierstrass substitution is primarily used for integrating rational functions of trigonometric functions. By substituting u = tan(x/2), it simplifies the integration process, allowing for easier manipulation of trigonometric identities. This method is particularly useful in calculus courses, where students encounter complex integrals involving sine, cosine, and other trigonometric functions.
How does the Weierstrass substitution simplify integrals?
The Weierstrass substitution transforms trigonometric functions into polynomial forms in terms of u. This allows integrals that would otherwise be difficult to solve directly to be approached using algebraic techniques, such as partial fractions. The resulting expressions for sin x, cos x, and other trigonometric functions become simpler, facilitating easier integration.
What are the key substitutions derived from the Weierstrass method?
Key substitutions derived from the Weierstrass method include sin x = 2u/(1 + u^2) and cos x = (1 - u^2)/(1 + u^2). Additionally, the method provides expressions for tan x, csc x, sec x, and cot x, all in terms of u. These substitutions are crucial for transforming integrals into a more manageable form.
Can you provide an example of using the Weierstrass substitution?
An example of using the Weierstrass substitution is the integral of sec x. By applying the substitution u = tan(x/2), the integral simplifies to a form that can be solved using partial fractions. The final result demonstrates the effectiveness of the Weierstrass method in finding integrals that involve secant and tangent functions.

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