u-Substitution and Inverse Trig Functions -...

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.