5.3. INTEGRATION BY SUBSTITUTION
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is linear, we can antidifferentiate composite functions according to the following rule.
If h(x) = f (ax + b) and F is a known algebraic antiderivative of f , then the general
antiderivative of h is given by
H(x) =
1
a
F(ax + b) + C.
When discussing antiderivatives, it is often useful to have shorthand notation that
indicates the instruction to find an antiderivative. Thus, in a similar way to how the
notation
d
dx
[ f (x)]
represents the derivative of f (x) with respect to x, we use the notation of the indefinite
integral,
f (x) dx
to represent the general antiderivative of f with respect to x. For instance, returning to
the earlier example with h(x) = (5x − 3) 6 above, we can rephrase the relationship between
h and its antiderivative H through the notation
(5x − 3)
6 dx =
1
35
(5x − 6)
7 + C.
When we find an antiderivative, we will often say that we evaluate an indefinite integral;
said differently, the instruction to evaluate an indefinite integral means to find the general
antiderivative. Just as the notation
d
dx [] means “find the derivative with respect to x of
,” the notation
dx means “find a function of x whose derivative is .”
Activity 5.7.
Evaluate each of the following indefinite integrals. Check each antiderivative that you
find by differentiating.
(a)
sin(8 − 3x) dx
(b)
sec 2 (4x) dx
(c)
1
11x−9 dx
(d)
csc(2x + 1) cot(2x + 1) dx
(e)
1
√
1−16x 2 dx
(f)
5 −x dx
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