Integration by Substitution: A Comprehensive Guide with Examples and Exercises, Slides of Differential and Integral Calculus

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ANTIDERIVATIVES

(INTEGRAL)

GENERALIZED POWER FORMULA

(Integration by Simple Substitution)

identify an integrand that can be

integrated using

simple substitution;

perform integration using the generalized

power

formula;

relate integration by power formula to the

generalized integration formula; and

consider and use the “introduction of

OBJECTIVES:

Or since F is an antiderivative of f,

For our purposes it will be useful to let

u=g(x) and to

write in the differential form

Thus,.



f (g(x))g'(x)dx  F(g(x))C

g' (x )

dx

du

 du  g'(x)dx



f (u)du  F(u)C

(1)

(2)

he generalized power formula therefore is:

   

 

; 1

1

( )

( ) ( )

1

  









C n

n

f u

f u d f u

n

n

The process of evaluating an integral of

the form (1) by converting it into the form

(2) with the substitution

is called

the method of u-substitution.

u  g( x) and du  g'(x) dx

EXERCISES

uate the integrals using appropriate substitut

rcises from page 338-339)

 

 





















dx

4 5 x

x

dx

5 x 2

x

dx

1 2 x

6

4. t 7 t 12 dt

5. ( 4 x 3 ) dx

2

3

3

3

2

9

 















a bx dx

dt

t

t

dx

x x

x

n

1

3

1

3

2