Unizor - Indefinite Integrals - Integration "By Parts"

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

Indefinite Integral -
Integration 'by Parts'
Examples

First, a reminder of integration 'by parts':
∫ [f(x) · g^I(x)] dx = f(x) · g(x) − ∫ [f^I(x) · g(x)] dx
Different form of this rule:
∫ f(x) · dg(x) = f(x) · g(x) − ∫ g(x) · df(x)
A short form can be written as:
∫ f·dg = f·g − ∫ g·df

Example 1:
∫ x·ln(x) dx 
Hint:
f(x)=ln(x) and
x·dx=dg(x)
Answer:
x²(2ln(x)−1)/4 + C

Example 2:
∫ e^x·cos(x) dx 
Hint:
Use integration 'by parts' twice.
Answer:
e^x(sin(x)+cos(x))/2 + C

Example 3:
∫ x√x+1 dx 
Hint:
u=x; dv=√x+1 dx;
v=(2/3)·(x+1)^3/2
Answer:
(2/3)·x·(x+1)^3/2 − (4/15)·(x+1)^5/2 + C

Example 4:
∫ x·ln²(x) dx 
Hint:
Integrate 'by parts' twice.Answer:
(1/2)x²ln²(x) − (1/2)x²ln(x) + (1/4)x²+C

Example 5:
∫ arctan(x) dx 
Answer:
x·arctan(x) − ln(1+x²)/2 + C

Example 6:
∫ x·arctan(x) dx 
Answer:
(x²+1)·arctan(x)/2 − x/2 + C