Integration Identifiers and Processes

Question 1

Describe how to Integrate:

∫ 1 / x²√(4 - x²) dx

Answer 1

Using Trig Substitution:

We first define the sides of the right triangle using the fact that:

a² + b² = c² and a = √[c² - b²] making:

a = √[c² - b²],
b = x
c = 2

Question 2

Describe how to Integrate:

∫ sin³θ dθ

Answer 2

Use Trig Identities along with U Substitution:

Change: ∫ sin³θ dθ → ∫ (1 - cos²θ)sinθ dθ

Now make U = cosθ, dU = - sinθ dθ

• ∫ (1 - u²) dU → ∫ u² - 1 dU

Then Integrate as a typical function subbing in cosθ where possible.

Question 3

Explain the process of solving:

Answer 3

Start by converting the sin⁵(11x) to a solveable form of [sin²11x]²sin11x

Then convert the inner sin²11x to [1-cos²11x] so that we can solve using u-sub

∫ [1-cos²11x]²sin(11x)dx → u = cos11x; du = -11sin11x

∫ [1 - u²]²du → ∫ [1 - u²]²du → ∫ [1 - 2u² + u⁴]du → u - 2u³/3 + u⁵/5

cos(11x) - 2cos³(11x)/3 + cos⁵(11x)/5 + C

Question 4

Answer 4

We want to convert cos(4x) into something more workable so we use cos(2*2x) and set u = 2x; du = 2dx

10(1/2) ∫ √[1 + cos2u] du

Then use the trig identity: cos2u = 2cos²u - 1

5 ∫ √[1 + (2cos²u - 1)] du → 5 ∫ √[2cos²u] du → 5 ∫ √[2]cosu du

5√[2]sin2x