Week 4 Lectures

[Pages:30]Week 4 Lectures

Janko Gravner MAT 21B

Jan. 24?28, 2022

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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.

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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

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8.2. Integration by parts

Example. Compute

x2 ln x dx

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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

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8.2. Integration by parts

Example. Compute

1

arctan x dx

0

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