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 half-angle formula to give an expression that is more easily integrable.
This is shown below.
Rewriting:
Use the half-angle formula to rewrite the integral:
sin(y)2
=
(1-cos(2y))/2.
This can be broken into separate integrals,
sin(
y)
2 dy
=
1/2
dy+
-(1/2)cos(2
y)
dy
which can be evaluated using
basic antidifferentiation, and
substitution (of "ax"),
respectively, to obtain
1/2
dy+
-(1/2)cos(2
y)
dy
=
((1/2)
y) + (-(1/4)sin(2
y)) +
C.
Explanation for rewritten term(s)
Basic antidifferentiationIn this case, by using
rules for basic
antiderivatives we can just write down the antiderivative of the
integrand:
1/2
dy
=
(1/2)
y +
C
Substitution (of "ax")Let
w = 2
y. Then
w' = 2, so
(1/2)
dw =
dy.
The integral can therefore be rewritten as
-(1/2)cos(2
y)
dy =
-(1/2)cos(
w) (1/2)
dw =
-(1/4)sin(
w) +
C
Thus, substituting back for
w,
-(1/2)cos(2
y)
dy =
-(1/4)sin(2
y) +
C
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