Lesson 2: Integrals Involving Trig Functions
Estimated time: 40-50 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Evaluate integrals of the form ∫ sinmx cosnx dx
- Use power-reduction identities for even powers of sine and cosine
- Evaluate integrals involving tan and sec
- Recognize when trig substitution is needed (introduction)
Key Trig Identities for Integration
Essential Identities
Pythagorean: sin²x + cos²x = 1, tan²x + 1 = sec²x
Half-angle (power-reduction):
sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2
Double angle: sin 2x = 2 sin x cos x
Strategy for ∫ sinmx cosnx dx
Decision Tree
- At least one odd power: Strip off one factor of sin x (or cos x), convert the remaining even powers using sin²x = 1 − cos²x (or cos²x = 1 − sin²x), then use u-substitution with u = cos x (or u = sin x).
- Both even powers: Use power-reduction (half-angle) identities to reduce the powers.
Example 1: Odd Power of Sine
Evaluate ∫ sin³x cos²x dx.
Step 1: sin³x = sin²x · sin x = (1 − cos²x) sin x.
Step 2: Let u = cos x, du = −sin x dx.
∫ (1 − u²) u² (−du) = −∫ (u² − u4) du = −u³/3 + u5/5 + C
= −cos³x/3 + cos5x/5 + C.
Example 2: Even Powers (Both Even)
Evaluate ∫ sin²x dx.
Step 1: Use power-reduction: sin²x = (1 − cos 2x)/2.
∫ (1 − cos 2x)/2 dx = x/2 − sin(2x)/4 + C = x/2 − sin(2x)/4 + C.
Example 3: cos²x sin²x
Evaluate ∫ cos²x sin²x dx.
Step 1: Use identity: cos x sin x = sin(2x)/2, so cos²x sin²x = sin²(2x)/4.
Step 2: Apply power-reduction: sin²(2x) = (1 − cos 4x)/2.
∫ (1 − cos 4x)/8 dx = x/8 − sin(4x)/32 + C.
Integrals Involving Tangent and Secant
Example 4: ∫ tan²x dx
Use the identity tan²x = sec²x − 1:
∫ (sec²x − 1) dx = tan x − x + C.
Example 5: ∫ sec³x dx (preview)
This integral requires integration by parts (covered in Lesson 3). The result is:
∫ sec³x dx = (1/2)[sec x tan x + ln|sec x + tan x|] + C.
This formula is worth memorizing as it appears frequently.
Products of Different Angles
For integrals like ∫ sin(mx) cos(nx) dx, use the product-to-sum formulas:
Product-to-Sum Formulas
sin A cos B = (1/2)[sin(A + B) + sin(A − B)]
sin A sin B = (1/2)[cos(A − B) − cos(A + B)]
cos A cos B = (1/2)[cos(A − B) + cos(A + B)]
Example 6: Product-to-Sum
Evaluate ∫ sin(3x) cos(5x) dx.
Step 1: = (1/2) ∫ [sin(8x) + sin(−2x)] dx = (1/2) ∫ [sin(8x) − sin(2x)] dx.
= (1/2)[−cos(8x)/8 + cos(2x)/2] + C = −cos(8x)/16 + cos(2x)/4 + C.
Introduction to Trig Substitution
Some integrals involve expressions like √(a² − x²), √(a² + x²), or √(x² − a²). These can be simplified with a trigonometric substitution:
Trig Substitution Guidelines
| Expression | Substitution | Identity Used |
|---|---|---|
| √(a² − x²) | x = a sin θ | 1 − sin²θ = cos²θ |
| √(a² + x²) | x = a tan θ | 1 + tan²θ = sec²θ |
| √(x² − a²) | x = a sec θ | sec²θ − 1 = tan²θ |
Trig Substitution Example
Example 7: Trig Substitution
Evaluate ∫ 1/√(4 − x²) dx.
Step 1: a = 2. Let x = 2 sin θ, dx = 2 cos θ dθ.
Step 2: √(4 − 4 sin²θ) = 2 cos θ.
Step 3: ∫ (2 cos θ)/(2 cos θ) dθ = ∫ dθ = θ + C = arcsin(x/2) + C.
Key Takeaways
- Odd power? Strip one factor, convert the rest using Pythagorean identity, u-sub.
- Both even? Use power-reduction (half-angle) identities.
- tan²x = sec²x − 1 is essential for tangent integrals.
- Trig substitution handles √(a² − x²), √(a² + x²), √(x² − a²) expressions.
Check Your Understanding
1. Evaluate ∫ cos³x dx.
2. Evaluate ∫ cos²x dx.
3. Evaluate ∫0π/2 sin5x dx.