This is an indefinite integral,
so we need to find
the most general
antiderivative of the integrand f(y).
To find an antiderivative of
f(
y), we go through
our list of integration methods:
- Recognize elementary antiderivatives
- Rewrite the integrand to make it easier
- Use substitution to reverse the chain rule or simplify the
integrand
- Use integration by parts
- Use inspection to see the value of a definite
integral
to find one that works for this integrand. In this case,
only one method is appropriate. We can rewrite the integrand: use the fact that cosine is an even function to give an expression that is more easily integrable.
This is shown below.
Rewriting:
Use the fact that cosine is an even function to rewrite the integral:
2y cos(-y)
=
2y cos(y).
Thus,
2
y cos(-
y)
dy
=
2
y cos(
y)
dy
which can be evaluated using
integration by parts, to obtain
2
y cos(
y)
dy
=
(2
y sin(
y) + 2 cos(
y)) +
C.
Explanation for rewritten term(s)
Integration by partsLet
u =
y and
v' = cos(
y).
Then
u' = 1 and
v = sin(
y),
so, remembering the integration by parts formula
u v'
dy =
u v -
u'
v dy,
we have
2
y cos(
y)
dy
=
(
y) (sin(
y)) -
(1) (sin(
y))
dy
=
2
y sin(
y) + 2 cos(
y) +
C.
![[ ]](/images/boxby.gif){kind=small}