1. 7-3 Integration by Parts

2. Remember:
*If we integrate both sides and rearrange it …
Integration by Parts Formula
“ultra-violet
voo-doo”
*Smart choosing of u and v will make the second integral easier to evaluate

3. Wouldn’t it be nice if we knew which way to choose them?
Here’s a helpful saying … L-I-A-T-E
L = log
I = inv trig
A = alg
T = trig
E = exp
Whichever comes first in the list …
let u =
… second in the list …
let dv =
One special mixed case:
e and sin/cos

4. Ex 1) Evaluate
Let u = x v = sin x
du = dx dv = cos x dx
Why do we have to do integration by parts in a certain order? Because won’t work, but will.
*Note: Integration by parts does not always work!

5. Ex 2) Evaluate (Repeated use of integration by parts)
Let u = x v = ex
du = 2x dx dv = ex dx
Let u = 2x v = ex
du = 2 dx dv = ex dx
(an easier way will be covered later today)

6. Confirm using slope field Y1 = x ln x Do Slopefield
Ex 3) Solve the differential equation subject to the initial
condition y = –1 when x = 1. Confirm the solution graphically by showing that it conforms to the slope field.
initial
(1, –1):
Confirm using slope field
Y1 = x ln x
Do Slopefield
Window [0, 5] x [–5, 5]
Graph Y2

7. Ex 4) Evaluate Let u = ex v = sin x du = ex dx dv = cos x dx
Let u = ex v = –cos x
du = ex dx dv = sin x dx
If you have to do it twice, keep choices for u & dv (don’t switch or you’ll undo)

8. f (x) and its derivatives g (x) and its antiderivatives
Tabular Integration
Used as a shortcut for integration by parts, if it meets certain criteria
* f (x) goes to 0 and * g (x) loops
f (x) and its derivatives
g (x) and its antiderivatives
(+)
(–)
(+)
etc.

9. Ex 5) Evaluate
(+)
(–)
(+)
(same as Ex 2 – compare answers )

10. Ex 6) Evaluate
(+)
(–)
(+)
(–)

11. homework
Pg. 350 #1 – 26 (skip mult of 3)