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) = ∫ f^I(x)dx = f(x) + C
Examples:
∫ dsin(x) = ∫ sin^I(x)dx =
= sin(x) + C
∫ cos(x)dx = ∫ sin^I(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 =
= ∫ 4x³dx + ∫ 3x²dx +
+ ∫ 2x³dx + ∫dx =
= x⁴ + x³ + x² + x + C

4. Integration "by-parts"
4a. ∫ [f(x) · g^I(x)] dx =
= f(x) · g(x) − ∫ [f^I(x) · g(x)] dx
Example:
∫ (x²·e^x) dx =
= x²·e^x − ∫ (2x·e^x) dx =
= x²·e^x − 2x·e^x + ∫ (2·e^x) dx =
= (x² − 2x + 2)·e^x + 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