8.2 integration by parts. start with the product rule:
TRANSCRIPT
8.2 Integration By Parts
Start with the product rule:
d dv duuv u v
dx dx dx
d uv u dv v du
d uv v du u dv
u dv d uv v du
u dv d uv v du
u dv d uv v du
u dv uv v du
u dv uv v du
The Integration by Parts formula is a “product rule” for integration.
u differentiates to zero (usually).
dv is easy to integrate.
Choose u in this order: LIPET
Logs, Inverse trig, Polynomial, Exponential, Trig
Integration by Parts
cos x x dxpolynomial factor u x
du dx
cos dv x dx
sinv x
u dv uv v du LIPET
sin cosx x x C
u v v du
sin sin x x x dx
Example
ln x dxlogarithmic factor lnu x
1du dx
x
dv dx
v x
u dv uv v du LIPET
lnx x x C
1ln x x x dx
x
u v v du
Example
This is still a product, so we need to use integration by parts again.
2 xx e dx u dv uv v du LIPET 2u x xdv e dx
2 du x dx xv e
u v v du 2 2 x xx e e x dx 2 2 x xx e xe dx
u x xdv e dx
du dx xv e 2 2x x xx e xe e dx 2 2 2x x xx e xe e C
Example
2 xx e dx & deriv.f x & integralsg x
2x
2x
2
0
xexexexe
2 xx e 2 xxe 2 xe C
Example
Tabular integration works for integrals of the form:
f x g x dx
where: Differentiates to 0 in several steps.
Integrates repeatedly.
A Shortcut: Tabular Integration
3 sin x x dx3 cosx x 2 3 sinx x 6 cosx x 6sin x + C
Example:
3x
23x
6x
6
sin x
cos x
sin xcos x
0
sin x
cos xe x dxxu e sin dv x dx
xdu e dx cosv x
u v v du sin sinx xe x x e dx
sin cos cos x x xe x e x x e dx
xu e cos dv x dx xdu e dx sinv x
sin cos cos x x xe x e x e x dx
This is the expression we started with!uv v du
cos xe x dx 2 cos sin cosx x xe x dx e x e x
sin coscos
2
x xx e x e xe x dx C
This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.
Example
• Try to choose u so that du (its derivative) becomes easier to integrate than u. – If ln is present, then u must be ln.– Oftentimes, let u be the powers of x.
• Also, choose dv so that it is easy to integrate dv.– If ex is present, let dv = ex dx– Oftentimes, let dv be the sin or cos.
• After integrating by parts, you should wind up with the integral that is “easier” to integrate.
How to choose u and dv
dxxx )2sin(
dxxx ln5 dxx)9arctan(
dxex x43
dxxe x sin25
1ln dtt
Examples