5.4. INTEGRATION BY PARTS
301
i. First, observe that
d
dx
[x sin(x)] = x cos(x) + sin(x).
Integrating both sides indefinitely and using the fact that the integral of a
sum is the sum of the integrals, we find that
d
dx
[x sin(x)]
dx =
x cos(x) dx +
sin(x) dx.
In this last equation, evaluate the indefinite integral on the left side as well
as the rightmost indefinite integral on the right.
ii. In the most recent equation from (i.), solve the equation for the expression
x cos(x) dx.
iii. For which product of basic functions have you now found the antiderivative?
⊲⊳
Reversing the Product Rule: Integration by Parts
Problem (c) in Preview Activity 5.4 provides a clue for how we develop the general technique
known as Integration by Parts, which comes from reversing the Product Rule. Recall that
the Product Rule states that
d
dx
[ f (x)g(x)] = f (x)g
′ (x) + g(x) f
′ (x).
Integrating both sides of this equation indefinitely with respect to x, it follows that
d
dx
[ f (x)g(x)] dx =
f (x)g
′ (x) dx +
g(x) f
′ (x) dx.
(5.6)
On the left in Equation (5.6), we recognize that we have the indefinite integral of the
derivative of a function which, up to an additional constant, is the original function itself.
Temporarily omitting the constant that may arise, we equivalently have
f (x)g(x) =
f (x)g
′ (x) dx +
g(x) f
′ (x) dx.
(5.7)
The most important thing to observe about Equation (5.7) is that it provides us with a
choice of two integrals to evaluate. That is, in a situation where we can identify two
functions f and g, if we can integrate f (x)g ′ (x), then we know the indefinite integral of
g(x) f ′ (x), and vice versa. To that end, we choose the first indefinite integral on the left in
301
i. First, observe that
d
dx
[x sin(x)] = x cos(x) + sin(x).
Integrating both sides indefinitely and using the fact that the integral of a
sum is the sum of the integrals, we find that
d
dx
[x sin(x)]
dx =
x cos(x) dx +
sin(x) dx.
In this last equation, evaluate the indefinite integral on the left side as well
as the rightmost indefinite integral on the right.
ii. In the most recent equation from (i.), solve the equation for the expression
x cos(x) dx.
iii. For which product of basic functions have you now found the antiderivative?
⊲⊳
Reversing the Product Rule: Integration by Parts
Problem (c) in Preview Activity 5.4 provides a clue for how we develop the general technique
known as Integration by Parts, which comes from reversing the Product Rule. Recall that
the Product Rule states that
d
dx
[ f (x)g(x)] = f (x)g
′ (x) + g(x) f
′ (x).
Integrating both sides of this equation indefinitely with respect to x, it follows that
d
dx
[ f (x)g(x)] dx =
f (x)g
′ (x) dx +
g(x) f
′ (x) dx.
(5.6)
On the left in Equation (5.6), we recognize that we have the indefinite integral of the
derivative of a function which, up to an additional constant, is the original function itself.
Temporarily omitting the constant that may arise, we equivalently have
f (x)g(x) =
f (x)g
′ (x) dx +
g(x) f
′ (x) dx.
(5.7)
The most important thing to observe about Equation (5.7) is that it provides us with a
choice of two integrals to evaluate. That is, in a situation where we can identify two
functions f and g, if we can integrate f (x)g ′ (x), then we know the indefinite integral of
g(x) f ′ (x), and vice versa. To that end, we choose the first indefinite integral on the left in