u-Substitution and Inverse Trig Functions -...
Transcript of u-Substitution and Inverse Trig Functions -...
![Page 1: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/1.jpg)
u-Substitution and Inverse Trig Functions
![Page 2: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/2.jpg)
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.
![Page 3: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/3.jpg)
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.
![Page 4: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/4.jpg)
When finished… Grab a white board and solve for “y”
2Solve y tdye
dt
Hint: Use a property of exponents
to separate the exponents
![Page 5: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/5.jpg)
HW Questions?---
TOMORROW TOMORROW!!
QUIZ
![Page 6: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/6.jpg)
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
![Page 7: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/7.jpg)
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
![Page 8: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/8.jpg)
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
![Page 9: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/9.jpg)
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
![Page 10: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/10.jpg)
Helpful Hint # 4 9ln x
dxx
4. If lnx is involved pick that for “u”
![Page 11: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/11.jpg)
In class practice . . . . Evaluate the following definite integrals.
1
2
0
3
4
4
1. 1
2. cot
r r dr
x dx
![Page 12: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/12.jpg)
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
![Page 13: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/13.jpg)
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
![Page 14: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/14.jpg)
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!
![Page 15: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/15.jpg)
u-Substitution with Inverse Trig Functions
![Page 16: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/16.jpg)
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
![Page 17: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/17.jpg)
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
![Page 18: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/18.jpg)
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
![Page 19: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/19.jpg)
More Practice!!!
2
2
2
4
402.
25
3.4
4.1 1
5. 1
dxx
dx
x
dx
x
tdt
t
![Page 20: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/20.jpg)
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
![Page 21: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/21.jpg)
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?
![Page 22: u-Substitution and Inverse Trig Functions - Weeblymurphymathematics.weebly.com/uploads/3/...5...trig.pdf · Inverse Trig Integrals 2 4 41 dx x What inverse trig function looks like](https://reader035.fdocuments.net/reader035/viewer/2022063006/5fb8175afcc90138e100c8d9/html5/thumbnails/22.jpg)
QUIZ Review – Finish Packet p. 7 Puzzle I would recommend you do the work on separate paper.