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

__Indefinite Integral -__

Integration 'by Parts'

Examples

Integration 'by Parts'

Examples

First, a reminder of integration 'by parts':

**∫****[**

*f(x) · g*]^{I}(x)*d*

**x = f(x) · g(x) − ∫****[**

*f*]^{I}(x) · g(x)*d*

**x**Different form of this rule:

**∫ f(x) ·**d**g(x) = f(x) · g(x) − ∫ g(x) ·**d**f(x)**A short form can be written as:

**∫ f·**d**g = f·g − ∫ g·**d**f**Example 1:

**∫ x·ln(x)**d**x***Hint:*

*and*

**f(x)=ln(x)**

**x·**d**x=**d**g(x)***Answer:*

**x²(2ln(x)−1)/4 + C**Example 2:

**∫ e**d^{x}·cos(x)**x***Hint:*

Use integration 'by parts' twice.

*Answer:*

**e**^{x}(sin(x)+cos(x))/2 + CExample 3:

**∫ x√x+1**d**x***Hint:*

**u=x; dv=√x+1**d**x;**

**v=(2/3)·(x+1)**^{3/2}*Answer:*

**(2/3)·x·(x+1)**^{3/2}− (4/15)·(x+1)^{5/2}+ CExample 4:

**∫ x·ln²(x)**d**x***Hint:*

Integrate 'by parts' twice.

*Answer:*

**(1/2)x²ln²(x) − (1/2)x²ln(x) + (1/4)x²+C**Example 5:

**∫ arctan(x)**d**x***Answer:*

**x·arctan(x) − ln(1+x²)/2 + C**Example 6:

**∫ x·arctan(x)**d**x***Answer:*

**(x²+1)·arctan(x)/2 − x/2 + C**
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