In this section, we will begin investigating
some more advanced techniques for
integration – specifically, integration by
parts.
Integration by Parts is essentially a product rule
for antidifferentiation.
It comes from the product rule from derivatives,
as we will see in the proof of the theorem.
If u and v are differentiable functions, then
1.
ò udv = uv - ò v du
2.
ò
b
a
udv = [uv] a - ò v du
b
b
a
dv should be selected so that v can be found by
antidifferentiating
Also, ò v du should be simpler to work with than
the original integral.
!!! Do not forget to consider a variable
substitution before using any more advanced
technique.
Choose u in this order:
L = logarithm
I = inverse trigonometry
A = algebraic functions
T = trigonometric functions
E = exponential functions
dv = rest of the original integrand
Find
3
x
ò ln x dx
Find
2x
xe
ò dx
Find
ò arcsin ( x) dx
Find
ò
p
p
2
4
x csc 2 x dx
Find
ò
x 2 e x dx
Find
x
e
ò cos (2x) dx
Find
ò
x 5 sin ( x 3 ) dx
Find
ò
xe- x dx
Find
2
x
ò arcsin ( x) dx