296
5.3. INTEGRATION BY SUBSTITUTION
From that perspective, we’ d have
5
2
xe
x 2 dx =
1
2
e
x 2
5
2
=
1
2
e
25 −
1
2
e
4
,
which is, of course, the same result.
Activity 5.9.
Evaluate each of the following definite integrals exactly through an appropriate usubstitution.
(a)
2
1
x
1 + 4x 2 dx
(b)
1
0
e
−x (2e
−x + 3)
9 dx
(c)
4/π
2/π
cos
1
x
x 2 dx
⊳
Summary
In this section, we encountered the following important ideas:
• To begin to find algebraic formulas for antiderivatives of more complicated algebraic
functions, we need to think carefully about how we can reverse known differentiation
rules. To that end, it is essential that we understand and recall known derivatives of
basic functions, as well as the standard derivative rules.
• The indefinite integral provides notation for antiderivatives. When we write “
f (x) dx,”
we mean “the general antiderivative of f .” In particular, if we have functions f and F
such that F ′ = f , the following two statements say the exact thing:
d
dx
[F(x)] = f (x) and
f (x) dx = F(x) + C.
That is, f is the derivative of F, and F is an antiderivative of f .
• The technique of u-substitution helps us evaluate indefinite integrals of the form
f (g(x))g ′ (x) dx through the substitutions u = g(x) and du = g ′ (x) dx, so that
f (g(x))g
′ (x) dx =
f (u) du.
5.3. INTEGRATION BY SUBSTITUTION
From that perspective, we’ d have
5
2
xe
x 2 dx =
1
2
e
x 2
5
2
=
1
2
e
25 −
1
2
e
4
,
which is, of course, the same result.
Activity 5.9.
Evaluate each of the following definite integrals exactly through an appropriate usubstitution.
(a)
2
1
x
1 + 4x 2 dx
(b)
1
0
e
−x (2e
−x + 3)
9 dx
(c)
4/π
2/π
cos
1
x
x 2 dx
⊳
Summary
In this section, we encountered the following important ideas:
• To begin to find algebraic formulas for antiderivatives of more complicated algebraic
functions, we need to think carefully about how we can reverse known differentiation
rules. To that end, it is essential that we understand and recall known derivatives of
basic functions, as well as the standard derivative rules.
• The indefinite integral provides notation for antiderivatives. When we write “
f (x) dx,”
we mean “the general antiderivative of f .” In particular, if we have functions f and F
such that F ′ = f , the following two statements say the exact thing:
d
dx
[F(x)] = f (x) and
f (x) dx = F(x) + C.
That is, f is the derivative of F, and F is an antiderivative of f .
• The technique of u-substitution helps us evaluate indefinite integrals of the form
f (g(x))g ′ (x) dx through the substitutions u = g(x) and du = g ′ (x) dx, so that
f (g(x))g
′ (x) dx =
f (u) du.