7.3 Trigonometric Substitution
Trigonometric Substitution
In finding the area of a circle or an ellipse, an integral of the form dx arises, where a > 0.
If it were the substitution u = a2 – x2 would be effective but, as it stands, dx is more difficult.
Trigonometric Substitution
If we change the variable from x to θ by the substitution x = a sin θ, then the identity 1 – sin2θ = cos2θ allows us to get rid of the root sign because
Trigonometric Substitution
Notice the difference between the substitution u = a2 – x2 (in which the new variable is a function of the old one) and the substitution x = a sinθ (the old variable is a function of the new one).
In general, we can make a substitution of the form x = g(t) by using the Substitution Rule in reverse.
To make our calculations simpler, we assume that g has an inverse function; that is, g is one-to-one.
Trigonometric Substitution
In this case, if we replace u by x and x by t in the Substitution Rule, we obtain
This kind of substitution is called inverse substitution.
We can make the inverse substitution x = a sin θ provided that it defines a one-to-one function.
Trigonometric Substitution
This can be accomplished by restricting θ to lie in the interval [–π/2, π/2].
In the following table we list trigonometric substitutions that are effective for the given radical expressions because of the specified trigonometric identities.
Trigonometric Substitution
In each case the restriction on θ is imposed to ensure that the function that defines the substitution is one-to-one.
Example 1
Evaluate Solution:
Let x = 3 sin θ, where –π/2 ≤ θ ≤ π/2. Then dx = 3 cos θ dθ and
(Note that cos θ ≥ 0 because –π/2 ≤ θ ≤ π/2.)
Example 1 – Solution
Thus the Inverse Substitution Rule gives
cont’d
Example 1 – Solution
Since this is an indefinite integral, we must return to the original variable x.
This can be done either by using trigonometric identities to express cot θ in terms of sin θ = x/3 or by drawing a diagram, as in Figure 1,
where θ is interpreted as an angle of a right triangle.
cont’d
sin θ =
Figure 1
Example 1 – Solution
Since sin θ = x/3, we label the opposite side and the hypotenuse as having lengths x and 3.
Then the Pythagorean Theorem gives the length of the adjacent side as so we can simply read the value of cot θ from the figure:
(Although θ > 0 in the diagram, this expression for cot θ is valid even when θ < 0.)
cont’d
Example 1 – Solution
Since sin θ = x/3, we have θ = sin–1(x/3) and so
cont’d
Example 2
Find the area enclosed by the ellipse
Solution:
Solving the equation of the ellipse for y, we get or
Example 2 – Solution
Because the ellipse is symmetric with respect to both axes, the total area A is four times the area in the first quadrant (see Figure 2).
cont’d
Figure 2
Example 2 – Solution
The part of the ellipse in the first quadrant is given by the function
and so
To evaluate this integral we substitute x = a sin θ. Then dx = a cos θ dθ.
cont’d
Example 2 – Solution
To change the limits of integration we note that when
x = 0, sin θ = 0, so θ = 0; when x = a, sin θ = 1, so θ = π/2.
Also
since 0 ≤ θ ≤ π/2.
cont’d
Example 2 – Solution
Therefore
cont’d
Example 2 – Solution
We have shown that the area of an ellipse with semiaxes a and b is πab.
In particular, taking a = b = r, we have proved the famous formula that the area of a circle with radius r is πr2.
cont’d
Trigonometric Substitution
Note:
Since the integral in Example 2 was a definite integral, we changed the limits of integration and did not have to
convert back to the original variable x.
Example 3
Find
Solution:
Let x = 2 tan θ, –π/2 < θ < π/2. Then dx = 2 sec2θ dθ and
=
= 2| sec θ|
= 2 sec θ
Example 3 – Solution
So we have
To evaluate this trigonometric integral we put everything in terms of sin θ and cos θ :
cont’d
Example 3 – Solution
Therefore, making the substitution u = sin θ, we have
cont’d
Example 3 – Solution
We use Figure 3 to determine that csc θ = and so
cont’d
Figure 3
Example 5
Evaluate where a > 0.
Solution 1:
We let x = a sec θ, where 0 < θ < π/2 or π < θ < 3π/2.
Then dx = a sec θ tan θ dθ and
Example 5 – Solution 1
Therefore
cont’d
Example 5 – Solution 1
The triangle in Figure 4 gives tan θ = so we have
cont’d
Figure 4
Example 5 – Solution 1
Writing C1 = C – In a, we have
cont’d
Example 5 – Solution 2
For x > 0 the hyperbolic substitution x = a cosh t can also be used.
Using the identity cosh2y – sinh2y = 1, we have
cont’d
Example 5 – Solution 2
Since dx = a sinh t dt, we obtain
Since cosh t = x/a, we have t = cosh–1(x/a) and
cont’d
Trigonometric Substitution
Note:
As Example 5 illustrates, hyperbolic substitutions
can be used in place of trigonometric substitutions and sometimes they lead to simpler answers.
But we usually use trigonometric substitutions because trigonometric identities are more familiar than hyperbolic identities.
Example 6
Find
Solution:
First we note that (4x2 + 9)3/2 = so trigonometric substitution is appropriate.
Although is not quite one of the expressions in the table of trigonometric substitutions, it becomes one of them if we make the preliminary substitution u = 2x.
Example 6 – Solution
When we combine this with the tangent substitution, we have x = which gives and
When x = 0, tan θ = 0, so θ = 0; when x = tan θ = so θ = π/3.
cont’d
Example 6 – Solution
Now we substitute u = cos θ so that du = –sin θ dθ. When θ = 0, u = 1; when θ = π/3, u =
cont’d
Example 6 – Solution
Therefore
cont’d
