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( sin ) .x x dx

2( sin )x x dx3

cos3

xx C

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.) b.) c.)( )f x dx ( )f u du ( )f u dx.) ( )a f x dx 3( 1)x dx

4

4

xx C

.) ( )b f u du 3( 1)u du 4

4

uu C

2 42( )

4

xx C

82

4

xx C

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.) b.) c.)( )f x dx ( )f u du ( )f u dx

( )f u dx 3( 1)u dx 2 3(( ) 1)x dx 6( 1)x dx

7

7

xx C

Using Substitution• Evaluate

• Let u = cos xdu/dx = -sin xdu = - 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