MAT01B1: Trig Substitution

Dr Craig

31 July 2018

My details:

I Dr Andrew Craig

I acraig@uj.ac.za

I Consulting hours:

Monday 14h40 – 15h25

Thursday 11h20 – 12h55

Friday 11h20 – 12h55

I Office C-Ring 508

https://andrewcraigmaths.wordpress.com/

or google “Andrew Craig maths”

General information

I Lectures:

Mon 15h30 – 17h05 (D-LAB K02)

Tue 08h50 – 10h25 (D-Les 101)

Mon & Tue lectures cover the same

topics.

Wed 17h10 – 18h45

(D-Les 101 and D-Les 102)

General information

I Tutorials:

Tue 13h50 – 15h25

(C-Les 101, D-Les 105)

Tue 15h30 – 17h05

(D-Les 101, D-Les 203)

I If you have a clash with the tutorial on a

Tuesday, please email me

(acraig@uj.ac.za).

General information

Saturday class this week:

D1 LAB 108

09h00 to 12h00

Focus on 7.1 – 7.4.

Tough integration by parts question:

Q21:

∫xe2x

(1 + 2x)2dx

Hint: let u = xe2x, dv =1

(1 + 2x)2dx.

Consider the function y =√1− x2.

This represents the top half the unit circle.

1−1

We want to calculate

∫ 1

−1

√1− x2 dx.

But first . . .

POP QUIZ:

Write down the following:

1. the formula for cos(A +B)

2. the three trig squared identities

3. a half-angle formula for cos2 x

4. the formula for sin(A +B)

5. a half-angle formula for sin2 x

6. a formula for sinA cosB

7. the formula for integration by parts

POP QUIZ ANSWERS:

1. cos(A +B) = cosA cosB − sinA sinB

2. sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ

1 + cot2 θ = csc2 θ

3. cos2 x = 12(1 + cos 2x)

4. sin(A +B) = sinA cosB + sinB cosA

5. sin2 x = 12(1− cos 2x)

6. sinA cosB = 12[sin(A−B)+ sin(A+B)]

7.∫u dv = uv −

∫v du

Remember now the example of∫ 1

−1

√1− x2 dx

Now for a > 0, consider the integral∫ √a2 − x2 dx

(This is a more general case of√1− x2.)

Now let x = a sin θ.

(θ ∈ [−π/2, π/2], more on this later)

Another example:

Consider the integral∫ √a2 + x2 dx

Here we will make a different substitution.

Let x = a tan θ, −π/2 < θ < π/2

For√x2 − a2 we use x = a sec θ.

(Restriction: θ ∈ [0, π/2) ∪ [π, 3π/2).)

Restrictions:

We need to restrict the allowable values of θ

when we make trig substitutions.

Because we are substituting, for example

x = a tan θ

we have to be sure that each value of θ will

produce a unique value for x. Therefore,

we must have that the trig function is

one-to-one on the interval that we allow.

Expression Substitution Restriction

√a2 − x2 x = a sin θ −π

26 θ 6

π

2

√a2 + x2 x = a tan θ −π

2< θ <

π

2

√x2 − a2 x = a sec θ 0 6 θ <

π

2or

π 6 θ <3π

2

Restrictions

The restrictions for trig substitutions must

always be included in your solution.

Also, they are very important when we do

definite integrals.

Example: ∫1

x2√x2 + 4

dx

Solution:

−√x2 + 4

4x+ C

Example:

Evaluate the integral∫ √9− x2x2

dx

Solution:

−√9− x2x

− arcsin(x3

)+ C

Original example: what about bounds?

∫ 1

−1

√1− x2 dx

We let x = sin θ, −π/2 6 θ 6 π/2.

If x = −1, then θ = −π/2.

If x = 1, then θ = π/2.

Note the importance of the restriction on θ.

Without this we would have (infinitely) many

options for θ when −1 = sin θ.

Example:

∫ 3√3/2

0

x3

(4x2 + 9)3/2dx

Solution: note that

(4x2 + 9)3/2 = (√4x2 + 9)3.

You might also need to substitute u = cos θ

at a later stage. Finally you will get

3

32

Example: ellipses

Find the area enclosed by the ellipse

x2

a2+y2

b2= 1

Firstly we see that

1

4A =

∫ a

0

b

a

√a2 − x2 dx

Example:

Evaluate ∫dx√x2 − a2

Solution:

`n|x +√x2 − a2| − `n|a| + C

= `n|x +√x2 − a2| + C1

(Note: hyperbolic trig substitution can also

be used for x > 0.)

Example: ∫x√x2 + 4

dx

Solution: use normal u-substitution to get

√x2 + 4 + C

Example with completing the square:∫x√

3− 2x− x2dx

Solution: complete the square to get

3− 2x− x2 = 4− (x + 1)2.

Substitute u = x + 1. Solution:

−√

3− 2x− x2 − arcsin

(x + 1

2

)+ C

Another example

∫ 2

0

√1 + 4x2 dx

Recall:

∫sec3 x dx =

1

2(secx tanx + ln | secx + tan x|) + C

Solution:∫ 2

0

√1 + 4x2dx =

1

4

(4√17 + ln(

√17 + 4)

)

Preparing for Chapter 7.4:

Practise your long division before Wednesday:

2x3 + 7x2 + 2x + 9

x2 + 3= ?

If you need more revision of long division,

watch the videos using the links provided in

the Blackboard announcement.