James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

472 Chapter 7 Techniques of Integration
Every differentiation rule has a corresponding integration rule. For instance, the Substitution
Rule for integration corresponds to the Chain Rule for differentiation. The rule
that corresponds to the Product Rule for differentiation is called the rule for integration
by parts.
The Product Rule states that if f and t are differentiable functions, then
d
f f sxdtsxdg − f sxdt9sxd 1 tsxd f 9sxd
dx
In the notation for indefinite integrals this equation becomes
y f f sxdt9sxd 1 tsxd f 9sxdg dx − f sxdtsxd
or
y f sxdt9sxd dx 1 y tsxd f 9sxd dx − f sxdtsxd
We can rearrange this equation as
1
y f sxdt9sxd dx − f sxdtsxd 2 y tsxd f 9sxd dx
Formula 1 is called the formula for integration by parts. It is perhaps easier to remember
in the following notation. Let u − f sxd and v − tsxd. Then the differentials are
du − f 9sxd dx and dv − t9sxd dx, so, by the Substitution Rule, the formula for integration
by parts becomes
2
y u dv − uv 2 y v du
Example 1 Find y x sin x dx.
SOLUTION USING FORMULA 1 Suppose we choose f sxd − x and t9sxd − sin x. Then
f 9sxd − 1 and tsxd − 2cos x. (For t we can choose any antiderivative of t9.) Thus,
using Formula 1, we have
y x sin x dx − f sxdtsxd 2 y tsxd f 9sxd dx
− xs2cos xd 2 y s2cos xd dx
− 2x cos x 1 y cos x dx
− 2x cos x 1 sin x 1 C
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