Week 4 Lectures
[Pages:30]Week 4 Lectures
Janko Gravner MAT 21B
Jan. 24?28, 2022
1
8.2. Integration by parts
Product formula:
(uv ) = u v + uv ,
or in differential form
d(uv ) = u dv + v du,
then rearranged
u dv = d(uv ) - v du,
and finally integrated
u dv = uv - v du.
8.2. Integration by parts
Integration by parts formula: u dv = uv - v du
This formula is useful when you integrate a product of two functions:
the first (u), simplifies by differentiation; and the second (dv ) does not get overly complicated by integration.
3
8.2. Integration by parts
Example. Compute
xe2x dx
8.2. Integration by parts
Take so that
xe2x dx = ()
u = x, du = dx dv = e2x dx , v = 1 e2x
2
() = x ? 1 e2x - 1 e2x dx
2
2
= 1 xe2x - 1 e2x + C
2
4
5
8.2. Integration by parts
Example. Compute
x2 ln x dx
6
8.2. Integration by parts
Example. Compute x2 ln x dx = ()
Take so that
1 u = ln x, du = dx
x dv = x2 dx, v = 1 x3
3
() = 1 x3 ? ln x - 1 x3 ? 1 dx
3
3x
= 1 x3 ? ln x - 1 x2 dx
3
3
= 1 x3 ? ln x - 1 x3 + C
3
9
8.2. Integration by parts
Example. Compute
1
arctan x dx
0
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