u-substitution and inverse trig functions -...
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u-Substitution and Inverse Trig Functions
Arrival
Pick a group of 3 people. (There may be TWO groups of 2..)
ONE member per group comes up to the front and picks a number from the Ms. Murphy.
Select ONE piece of construction paper per group and write all names at the top of the paper.
Activity (packet pg. 7) Work on the problem that corresponds to the # you drew)
No Packet? Here are the problems . . .
41. (3 1)x dx
2
2. xxe dx
23. 9x x dx
2 3
24.
( 4 3)
xdx
x x
35. 4(3 1)x dx
2 36. 4 (9 )x x dx
2 37. 18 cos(3 )x x dx
3 48. 12 sin(3 )x x dx
THEN… Show your work to Ms. Murphy!
Once approved, you may hang up your
paper in the back of the room.
When finished… Grab a white board and solve for “y”
2Solve y tdye
dt
Hint: Use a property of exponents
to separate the exponents
HW Questions?---
TOMORROW TOMORROW!!
QUIZ
Identify the “u” for the most effective
u-substitution for integrating
Find antiderivatives by substitution of variables
Integrate inverse trig functions using u-substitution
Evaluate definite integrals using u-substitution
Practice sintan
cos
xx dx dx
x
What should we pick for ?
sin OR cos
u
u x u x
cos du x dx sin
sin
du x dx
du x dx
1We would need .
cos
Not a good pick for u.
x
1du
u
ln ln cosu C x C
U-Substitution—More About Picking u Helpful Hints #1 and #2
1. Try to find which factor
is close to being the
derivative of the other
2. The higher power is often u
2 3( 5) (2 ) x x dx
2therefore, 5u x
2 5 2d
x xdx
sin3 xcos x dx
3rd power 1st power
therefore, sinu x
Helpful Hint # 3
sin
cos
xdx
xcosu x
sin
sin
du x dx
du x dx
3. Try letting the denominator be u
Check to see that the numerator is du
Helpful Hint # 4 9ln x
dxx
4. If lnx is involved pick that for “u”
In class practice . . . . Evaluate the following definite integrals.
1
2
0
3
4
4
1. 1
2. cot
r r dr
x dx
Here’s a tricky one:
x 5x 2 dxu x5 2du dx5
1
5du dx
u x5 2
Use original:
u x2 51
52( )u x
x 5x 2 dx
1
52( )u u
1
2 1
5du
Solve, but don’t forget to substitute back in the x’s!
1
25(u 2)u
1
2du
1
52( )u u
1
2 1
5du
1
25(u
3
2 2u1
2 )du
Final steps should be:
1
25
2
5
4
3
5
2
3
2[ ]u u C1
25(u
3
2 2u1
2 )du
1
25
2
55 2
4
35 2
5
2
3
2[ ( ) ( ) ]x x C
1
255 2
2
55 2
4
3
3
2( ) [ ( ) ]x x C
Lovely!
u-Substitution with Inverse Trig Functions
Try to Solve using u-Substitution
2
4
4 1dx
x u 4x2 1
du 8x dx
There is no extra x in the original
to substitute for
Enter Inverse Trig Functions You need to remember the derivatives of the inverse trig functions:
1
1
1
1
1
1
cos
sin
tan
csc
sec
cot
d u
dx
d u
dx
d u
dx
d u
dx
d u
dx
d u
dx
Inverse Trig Integrals
2
4
4 1dx
xWhat inverse trig function looks like the
best fit?
Get the 1 in the bottom
Factor constants outside the integral
Write as (something)2
Use u-Substitution with inverse trig formula
More Practice!!!
2
2
2
4
402.
25
3.4
4.1 1
5. 1
dxx
dx
x
dx
x
tdt
t
More Practice Answers 1
2
1
2
1
2
1 2
4
402. 8 tan
25 5
3. sin24
4. sin 11 1
15. sin
21
xC
x
dx xC
x
dxx C
x
tdt t C
t
Identify the “u” for the most effective
u-substitution for integrating
Find antiderivatives by substitution of variables
Integrate inverse trig functions using u-substitution
Evaluate definite integrals using u-substitution
Thumbs up?
QUIZ Review – Finish Packet p. 7 Puzzle I would recommend you do the work on separate paper.