Transcript Integration by parts

Antidifferentiation by Substitution
• If y = f(x) we can denote the derivative of f by either
dy/dx or f’(x). What can we use to denote the
antiderivative of f?
– We have seen that the general solution to the
differential equation dy/dx = f(x) actually consists of an
infinite family of functions of the form F(x) + C, where
F’(x) = f(x).
• Both the name for this family of functions and the symbol we
use to denote it are closely related to the definite integral
because of the Fundamental Theorem of Calculus.
• The symbol is an integral sign, the function f is

the integrand of the integral, and x is the variable of
integration.
Evaluating an Indefinite Integral
• Evaluate

2
(
x

sin
x
)
dx
.

3
x
( x  sin x)dx   cos x  C
3
2
Paying Attention to the Differential
• Let f(x) = x³ + 1 and let u = x². Find each of the
following antiderivatives in terms of x:
a.) f ( x)dx b.)
f (u )du c.) f (u )dx



4
x
3
a.) f ( x)dx   ( x  1)dx   x  C
4

4
u
b.)  f (u )du   (u  1)du   u  C
4
2 4
8
(x )
x
2

 x  C   x2  C
4
4
3
Paying Attention to the Differential
• Let f(x) = x³ + 1 and let u = x². Find each of the
following antiderivatives in terms of x:
a.) f ( x)dx b.)
f (u )du c.) f (u )dx




f (u )dx   (u 3  1)dx   (( x2 )3  1)dx
7
x
 ( x  1)dx   x  C
7

6
Using Substitution
• Evaluate
• Let u = cos x
du/dx = -sin x
du = - sin x dx
Using Substitution
• Evaluate
• Let u = 5 + 2x³, du = 6x² dx.
Using Substitution
• Evaluate
• We do not recall a function whose derivative is
cot 7x, but a basic trigonometric identity changes the
integrand into a form that invites the substitution u = sin
7x, du = 7 cos 7x dx.
Setting Up a Substitution with a Trigonometric Identity
• Find the indefinite integrals. In each case you can
use a trigonometric identity to set up a substitution.
Setting Up a Substitution with a Trigonometric Identity
• Find the indefinite integrals. In each case you can
use a trigonometric identity to set up a substitution.
Setting Up a Substitution with a Trigonometric Identity
• Find the indefinite integrals. In each case you can
use a trigonometric identity to set up a substitution.
Evaluating a Definite Integral by Substitution
• Evaluate
• Let u = tan x and du = sec²x dx.
That Absolute Value Again
• Evaluate