## Monday, February 13, 2017

### Unizor - Indefinite Integrals - Basic Properties

Notes to a video lecture on http://www.unizor.com

Indefinite Integral -
Basic Properties

1. Integral of differential
df(x) =  fI(x)dx = f(x) + C
Examples:
dsin(x) =  sinI(x)dx =
= sin(x) + C

cos(x)dx =  sinI(x)dx =
= sin(x) + C

2. Constant multiplier
a·f(x)dx = a· f(x)dx
Example:
5·x4dx = 5· x4dx =
= 5·x5/5 = x5

3. Sum of functions
[f(x) + g(x)] dx =
f(x)
dx +  g(x)dx

Example:
(4x³+3x²+2x+1) dx =
4x³
dx +  3x²dx +
2x³
dx + dx =
= x4 + x³ + x² + x + C

4. Integration "by-parts"
4a.  [f(x) · gI(x)] dx =
= f(x) · g(x) −
[fI(x) · g(x)] dx
Example:
(x²·exdx =
= x²·ex −  (2x·ex
dx =
= x²·ex − 2x·ex +  (2·ex
dx =
= (x² − 2x + 2)·ex + C

4b.  f(x) · dg(x) =
= f(x) · g(x) −  g(x) ·
df(x)

Example:
x² dsin(x) =
= x²·sin(x) −  sin(x)
d(x²) =
= x²·sin(x) −  2x·sin(x)
dx =
= x²·sin(x) +  2x
dcos(x) =
= x²·sin(x) + 2x·cos(x) −
− 2sin(x) + C

4c.  f(x) dx = f(x)·x −  x df(x)
Example:
ln(x) dx =
= x·ln(x) −  x
dln(x) =
= x·ln(x) −  x·(1/x)
dx =
= x·ln(x) − x + C