Trigonometric Integrals

Key Trigonometric Formulas

• Pythagorean identities:
  • \sin^2(x) + \cos^2(x) = 1
  • 1 + \tan^2(x) = \sec^2(x)
  • 1 + \cot^2(x) = \csc^2(x)
• Double angle formulas:
  • \sin(2x) = 2 \sin(x) \cos(x)
  • \cos(2x) has three forms:
    • \cos^2(x) - \sin^2(x)
    • 1 - 2 \sin^2(x)
    • 2 \cos^2(x) - 1
• Power-reducing formulas:
  • \sin^2(x) = \frac{1}{2} (1 - \cos(2x))
  • \cos^2(x) = \frac{1}{2} (1 + \cos(2x))

Integrals of Specific Trigonometric Functions

Example 1: Integral of \cos^3(x)

1. Rewrite: \cos^3(x) = \cos^2(x) \cdot \cos(x)
2. Use identity: \cos^2(x) = 1 - \sin^2(x)
3. Substitution:
   • Let u = \sin(x),
   • Then, du = \cos(x) dx
4. Resulting integral: \int (1 - u^2) du
5. Antiderivative:
   • u - \frac{u^3}{3} + C
6. Substitute back: \sin(x) - \frac{\sin^3(x)}{3} + C

Example 2: Integral of \cos^5(x)

1. Rewrite: \cos^5(x) = \cos^4(x) \cdot \cos(x)
2. Use identity: \cos^4(x) = (\cos^2(x))^2 = (1 - \sin^2(x))^2
3. Substitution: u = \sin(x), du = \cos(x) dx
4. Expand: (1 - u^2)^2 = 1 - 2u^2 + u^4
5. Resulting integral: \int (1 - 2u^2 + u^4) du
6. Antiderivative: u - \frac{2u^3}{3} + \frac{u^5}{5} + C
7. Substitute back: \sin(x) - \frac{2\sin^3(x)}{3} + \frac{\sin^5(x)}{5} + C

Example 3: Integral of \cos^5(x) \cdot \sin(x)

1. Substitution: u = \cos(x), du = -\sin(x) dx
2. Resulting integral: -\int u^5 du
3. Antiderivative: -\frac{u^6}{6} + C
4. Substitute back: -\frac{\cos^6(x)}{6} + C

Example 4: Integral of \sin^5(x) \cos^2(x)

1. Rewrite: \sin^5(x) = (\sin^4(x))\sin(x), \cos^2(x) is left.
2. Identity: \sin^4(x) = (\sin^2(x))^2 and \sin^2(x) = 1 - \cos^2(x)
3. Substitution: u = \cos(x), du = -\sin(x) dx
4. Expand and integrate: \int (1 - u^2)^2 u^2 du
5. Antiderivative: - (\frac{u^3}{3} - \frac{2u^5}{5} + \frac{u^7}{7}) + C
6. Substitute back: - (\frac{\cos^3(x)}{3} - \frac{2\cos^5(x)}{5} + \frac{\cos^7(x)}{7}) + C

Example 5: Integral of \sin^2(x)

1. Use power-reducing formula: \sin^2(x) = \frac{1}{2}(1 - \cos(2x))
2. Resulting integral: \frac{1}{2} \int (1 - \cos(2x)) dx
3. Antiderivative: \frac{1}{2} (x - \frac{\sin(2x)}{2}) + C

Example 6: Integral of \cos^2(3x)

1. Use power-reducing formula: \cos^2(x) = \frac{1}{2}(1 + \cos(2x))
2. Adjustment for angle: \cos^2(3x) = \frac{1}{2}(1 + \cos(6x))
3. Resulting integral: \frac{1}{2} \int (1 + \cos(6x)) dx
4. Antiderivative: \frac{1}{2} (x + \frac{\sin(6x)}{6}) + C

Example 7: Integral of \sin^4(x)

1. Use power-reducing formula: \sin^2(x) = \frac{1}{2}(1 - \cos(2x)) and square it.
2. Expansion and integration steps similar to previous examples.

Example 8: Integral of \sin^2(x) \cos^2(x)

1. Use power-reducing formula for both sine and cosine.
2. Resulting integral: \frac{1}{4} \int (1 - \cos(2x))(1 + \cos(2x)) dx
3. Simplify and integrate.

Example 9: Integral of \cos^2(x) \tan^3(x)

1. Convert tangent to sine and cosine and rewrite.
2. Simplify and integrate using substitution.

Summary

• Many trigonometric integrals can be solved using identities and appropriate substitutions.
• Familiarity with trigonometric identities and power-reducing formulas is crucial.
• Always look to simplify integrands using known identities before integrating.