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

__Indefinite Integral -__

Basic Properties

Basic Properties

1. Integral of differential

**∫**d**f(x) = ∫ f**d^{I}(x)**x = f(x) + C**Examples:

**∫**d**sin(x) = ∫ sin**d^{I}(x)**x =**

= sin(x) + C= sin(x) + C

**∫ cos(x)**d**x = ∫ sin**d^{I}(x)**x =**

= sin(x) + C= sin(x) + C

2. Constant multiplier

**∫ a·f(x)**d**x = a·∫ f(x)**d**x**Example:

**∫ 5·x**d^{4}**x = 5·∫ x**d^{4}**x =**

= 5·x= 5·x

^{5}/5 = x^{5}3. Sum of functions

**∫****[**

*f(x) + g(x)*]*d*

**x =**

= ∫ f(x)d= ∫ f(x)

**x + ∫ g(x)**d**x**Example:

**∫****(**

*4x³+3x²+2x+1*)*d*

**x =**

= ∫ 4x³d= ∫ 4x³

**x + ∫ 3x²**d**x +**

+ ∫ 2x³d+ ∫ 2x³

**x + ∫**d**x =**

= x= x

^{4}+ x³ + x² + x + C4. Integration "by-parts"

4a.

**∫****[**

*f(x) · g*]^{I}(x)*d*

**x =**

= f(x) · g(x) − ∫= f(x) · g(x) − ∫

**[**

*f*]^{I}(x) · g(x)*d*

**x**Example:

**∫ (x²·e**d^{x})**x =**

= x²·ed= x²·e

^{x}− ∫ (2x·e^{x})**x =**

= x²·ed= x²·e

^{x}− 2x·e^{x}+ ∫ (2·e^{x})**x =**

= (x² − 2x + 2)·e= (x² − 2x + 2)·e

^{x}+ C4b.

**∫ f(x) ·**d**g(x) =**

= f(x) · g(x) − ∫ g(x) ·d= f(x) · g(x) − ∫ g(x) ·

**f(x)**Example:

**∫ x²**d**sin(x) =**

= x²·sin(x) − ∫ sin(x)d= x²·sin(x) − ∫ sin(x)

**(x²) =**

= x²·sin(x) − ∫ 2x·sin(x)d= x²·sin(x) − ∫ 2x·sin(x)

**x =**

= x²·sin(x) + ∫ 2xd= x²·sin(x) + ∫ 2x

**cos(x) =**

= x²·sin(x) + 2x·cos(x) −

− 2sin(x) + C= x²·sin(x) + 2x·cos(x) −

− 2sin(x) + C

4c.

**∫ f(x)**d**x = f(x)·x − ∫ x**d**f(x)**Example:

**∫ ln(x)**d**x =**

= x·ln(x) − ∫ xd= x·ln(x) − ∫ x

**ln(x) =**

= x·ln(x) − ∫ x·(1/x)d= x·ln(x) − ∫ x·(1/x)

**x =**

= x·ln(x) − x + C= x·ln(x) − x + C

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