📐 Concept diagram

05-06 - Trigonometric Integrals

Phase: 5 | Subject: 05-06 Prerequisites: 05-05-integration-by-parts.md Next subject: 05-07-partial-fractions-integration.md

Learning Objectives

By the end of this subject, you will be able to:

1. Integrate sinᵐx·cosⁿx using appropriate strategies
2. Integrate tanᵐx·secⁿx
3. Apply half-angle formulas for difficult integrals
4. Use product-to-sum formulas

Core Content

Strategy for sinᵐx·cosⁿx

Case 1: n is odd

Save one cos(x), convert rest to sin(x) using cos²x = 1 - sin²x. ∫sinᵐx·cosⁿxdx = ∫sinᵐx·cosⁿ⁻¹x·cos(x)dx

Case 2: m is odd

Save one sin(x), convert rest to cos(x) using sin²x = 1 - cos²x. ∫sinᵐx·cosⁿxdx = ∫sinᵐ⁻¹x·cosⁿx·sin(x)dx

Case 3: Both m and n even

Use half-angle formulas: sin²x = (1 - cos(2x))/2 cos²x = (1 + cos(2x))/2

Strategy for tanᵐx·secⁿx

Case 1: n is even

Save sec²x, convert rest to tan(x) using sec²x = 1 + tan²x. ∫tanᵐx·secⁿxdx = ∫tanᵐx·secⁿ⁻²x·sec²xdx

Case 2: m is odd

Save sec(x)·tan(x), convert rest to sec(x) using tan²x = sec²x - 1. ∫tanᵐx·secⁿxdx = ∫tanᵐ⁻¹x·secⁿ⁻¹x·sec(x)·tan(x)dx

Half-Angle Formulas

$sin²(x) = (1 - cos(2x))/2
cos²(x) = (1 + cos(2x))/2
sin(x)·cos(x) = (1/2)sin(2x)
$

Product-to-Sum

$sin(A)·sin(B) = (1/2)[cos(A-B) - cos(A+B)]
cos(A)·cos(B) = (1/2)[cos(A-B) + cos(A+B)]
sin(A)·cos(B) = (1/2)[sin(A-B) + sin(A+B)]
$

Key Terms

• 05 06 Trigonometric Integrals
• Case 1: n is even
• Case 1: n is odd
• Case 2: m is odd
• Case 3: Both m and n even
• Correct: A)
• Correct: B)
• Example 1: Odd power of cos
• Example 2: Even powers
• Example 3: tan and sec
• Half-Angle Formulas
• Product-to-Sum

Worked Examples

Example 1: Odd power of cos

∫sin²x·cos³xdx = ∫sin²x·cos²x·cos(x)dx = ∫sin²x(1 - sin²x)·cos(x)dx Let u = sin(x), du = cos(x)dx = ∫u²(1 - u²)du = ∫(u² - u⁴)du = u³/3 - u⁵/5 + C = sin³(x)/3 - sin⁵(x)/5 + C

Example 2: Even powers

∫sin²xdx = ∫(1 - cos(2x))/2 dx = (1/2)∫dx - (1/2)∫cos(2x)dx = x/2 - (1/4)sin(2x) + C

Example 3: tan and sec

∫tan³x·sec(x)dx = ∫tan²x·sec(x)·tan(x)dx = ∫(sec²x - 1)·sec(x)·tan(x)dx Let u = sec(x), du = sec(x)·tan(x)dx = ∫(u² - 1)du = u³/3 - u + C = sec³(x)/3 - sec(x) + C

Practice Problems

Problem 1: ∫cos³xdx

Problem 2: ∫sin²x·cos²xdx

Problem 3: ∫tan²x·sec(x)dx

Problem 4: ∫sin(3x)·cos(2x)dx

Problem 5: ∫cos⁴x dx

Summary

Key takeaways:

• sinᵐ·cosⁿ: use odd power strategy or half-angle formulas
• tanᵐ·secⁿ: save sec²x or sec·tan depending on which power is odd
• Half-angle: sin² = (1-cos2x)/2, cos² = (1+cos2x)/2
• Product-to-sum converts products to sums

Pitfalls

• Applying the wrong strategy for the parity. For ∫sinᵐx·cosⁿx, the strategy depends on which powers are odd. Trying to save a cosine factor when the sine power is odd (or vice versa) leads to integrals that don't simplify under u-substitution.
• Forgetting the factor of 1/2 in half-angle formulas. sin²x = (1 − cos(2x))/2. Students often drop the 1/2 and integrate (1 − cos(2x)) instead, doubling the answer. The same applies to cos²x.
• Not recognizing when product-to-sum is needed. Integrals like ∫sin(3x)·cos(2x)dx look like sin·cos but the arguments differ — half-angle identities don't apply here. Product-to-sum formulas are the correct tool.
• Misidentifying tan·sec strategies. For ∫tanᵐx·secⁿx, saving sec²x works when n is even; saving sec(x)·tan(x) works when m is odd. Confusing these two cases — e.g., trying to save sec²x when m is odd and n is odd — doesn't produce a useful substitution.
• Losing terms when repeatedly applying half-angle. For ∫cos⁴x dx, applying cos²x = (1+cos(2x))/2 twice requires careful algebra. The cross-term 2cos(2x) from squaring is frequently dropped, leading to missing terms in the final answer.

Quiz

Q1: ∫sin²xdx equals:

A) -cos(x) + C B) x/2 - sin(2x)/4 + C C) -cos²(x) + C D) sin²(x)/2 + C

Q2: For ∫sin³x·cos²xdx, the best strategy is:

A) Save one sin(x), convert rest to cos B) Save one cos(x), convert rest to sin C) Use half-angle on both D) Use substitution u = tan(x)

Q3: ∫cos²x·sin²xdx equals:

A) x/2 + C B) x/8 - sin(4x)/32 + C C) sin(4x)/8 + C D) (x - sin(x)·cos(x))/2 + C

Q4: ∫tan²x·sec²xdx equals:

A) tan³x/3 + C B) tan(x) + C C) sec²(x) + C D) tan²x + C

Q5: The half-angle formula for sin²x is:

A) (1 + cos(2x))/2 B) (1 - cos(2x))/2 C) cos(2x) D) 1 - cos²(x)

Next Steps

Next up: 05-07-partial-fractions-integration.md