This is an indefinite integral,
so we need to find
the most general
antiderivative of the integrand f(x).
To find an antiderivative of
f(
x), 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: consider the parts of the integral separately to give an expression that is more easily integrable.
This is shown below.
Rewriting:
Consider the parts of the integral separately. This can be broken into separate integrals,
6
e5x - 5
e6x dx
=
6
e5xdx+
-5
e6xdx
which can be evaluated using
substitution (of "ax"), and
substitution (of "ax"),
respectively, to obtain
6
e5xdx+
-5
e6xdx
=
((6/5)
e5x) + (-(5/6)
e6x) +
C.
Explanation for rewritten term(s)
Substitution (of "ax")Let
w = 5
x. Then
w' = 5, so
(1/5)
dw =
dx.
The integral can therefore be rewritten as
6
e5x dx =
6
ew (1/5)
dw =
(6/5)
ew +
C
Thus, substituting back for
w,
6
e5x dx =
(6/5)
e5x +
C
Substitution (of "ax")Let
w = 6
x. Then
w' = 6, so
(1/6)
dw =
dx.
The integral can therefore be rewritten as
-5
e6x dx =
-5
ew (1/6)
dw =
-(5/6)
ew +
C
Thus, substituting back for
w,
-5
e6x dx =
-(5/6)
e6x +
C
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