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Detailed Contents
Index
Programmed tutorial: Integral of Cosine Cubed
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Let’s find the indefinite integral
Equation 1
General Contents
Detailed Contents
Index
Can the problem be converted to a more convenient form?
Possibly it can if we substitute for (6
x
2
- 8
x
- 5).
Try the substitution (6
x
2
- 8
x
- 5) =
u
.
What is needed for
du
?
In Equation 1 we have only (2
x
– 3)*
dx
.
Express it in terms
of
du
.
Express Equation 1 in terms
of
u
and
du
?
Equation 2
Do we have integrals for any power of cosine
u
?
Yes, for the first power.
State that integral.
For an odd power of the cosine, like the power 3 in Equation 2, it helps to separate off one power of the cosine and replace the remaining powers.
Here that means writing
Try replacing the cos
2
u
by using the Pythagorean Identity.
We get
Distribute the multiplication and write as two integrals.
Do you see how to integrate the first integral?
Yes, as discussed above.
For the second integral,
let’s try the substitution
w
=
sin
(
u
).
Find the
dw
required.
Rewrite
in terms of
w
and
dw
.
Equation.3
Integrate to get an expression in terms of
w
.
We can use the power rule:
Use
w
=
sin
(
u
) to rewrite this in terms of
u
.
We get
Use these results to evaluate Equation 3.
So we see that
can be integrated.
Combine all of these results to find
I
1
in terms of
u
.
Rewrite this result in terms of
x
.
This is our final result.
How can we check our work?
We can differentiate this result to see if we get the integrand in Equation 1.
Do the differentiation, remembering to use the Chain Rule.
Replace sin
2
in the second line by 1 – cos
2
.
We get
Distribute the multiplication in the second line.
The first two terms eliminate, giving
which checks.
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Equation 1