5. Indefinite Integrals

Calculus 2



Weiqi Zhou

Anti-derivatives

if on some interval
then is an anti-derivative of on

The Family of Anti-derivatives

  • apparently if is an anti-derivative of on ，then so is for any

  • conversely if are anti-derivatives of on ，then for some

  • is the family of anti-derivatives of on

Indefinite Integral

the map
is called the indefinite integral of on ，and denoted by

Example







Memortize the Table of Integrals

Linearity

Computations I

Example

  • find

  • let ，then

Example

  • find

  • let ，then

Exercises

  • find following indefinite integrals





Computations II

Example

  • find

  • let ，then



Substitution Techniques

  • take if the integrand contains ，

  • take if the integrand contains ，

  • take if the integrand contains ，

Example

  • find

  • let ，then



Exercises

  • find following indefinite integrals



Integration by Parts

Example

  • find

  • take ,



Example

  • find

  • take ,




Example

  • find

  • take ,


Typical Cases

  • the order of the integrand can be lowered by differentiation

  • the integrand contains trignomeric or exponential functions

  • the integrand can be simplified or converted to other cases mentioned above

Exercises

  • find following indefinite integrals





Summary

  • Indefinite Integrals

  • Computations of Indefinite Integrals

  • Substitutions

  • Integration by Parts