Tabular Integration
Tabular Integration can be used as a viable substitute for Integration by Parts when you have a terminating u term.
Take the following as an example:
∫x e
3 2x
dx .
When doing integration by parts
u = x3
and
dv = e 2 x dx
We will use these two terms in our tabular integration
Differentiate
Integrate
x3
e2x
1 2x
e
2
1 2x
e
4
1 2x
e
8
1 2x
e
16
3x2
6x
6
0
Now you multiply down the diagonals alternating
signs, starting with + (then –) so you have
x3
1 2x
1
1
1
e - 3x2 e2x + 6x e2x - 6 e2x + C
2
4
8
16
1
3
3
3
= x3e2x - x2e2x + xe2x - e2x + C
2
4
4
8
Check: Using Integration by Parts
3 2x
∫ x e dx
u = x3
and
dv = e 2 x dx
∫
1 3 2x 3 x 2 e 2 x dx
u = x 2 and
xe 2
2
2x
1 3 2x 3 1 2 2x
x e - [ x e - xe dx ] =
2
2 2
so
du = 3x 2 dx
dv = e 2 x dx
so
and
v=
1 2x
e
2
du = 2 x dx
and
v=
1 2x
e
2
∫
1 3 2x 3
xe 2
4
1 3 2x 3
xe 2
4
1 3 2x 3
xe 2
4
1 3 2x 3
xe 2
4
∫
3 xe 2 x dx
u = x and dv = e 2 x dx so du = dx and v =
2
3 1
1 e 2 x dx
x2e2x + [ xe2x ]=
2 2
2
3
3 e 2 x dx
1 3 2x 3 2 2x 3 2x 3 1 2x
x2e2x + xe2x =
x e - x e + xe - [ e ] +C =
4
4
2
4
4
4 2
3
3
x2e2x + xe2x - e2x + C
4
8
x2e2x +
1 2x
e
2
∫
∫
Our result with tabular integration above matches the result with integration by parts.
This handout is available at: www.uvu.edu/mathlab/handouts.html