Trigonometric Integrals Lecture Notes

Useful Trigonometric Identities

• **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:"

  • $\sin(2x) = 2 \sin(x) \cos(x)$
  • $\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1$

• **Power Reducing Formulas:"

  • $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$
  • $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$

Examples and Solutions

Anti-derivative of $\cos^3(x)$

1. Start with $\cos^2(x) \cos(x)$.
2. Use $\cos^2(x) = 1 - \sin^2(x)$.
3. Substitute $u = \sin(x)$, $du = \cos(x) \ dx$.
4. Integral becomes $\int (1 - u^2) du = u - \frac{u^3}{3} + C$
5. Substitute back $u = \sin(x)$ for the final answer:
   • $\boxed{\sin(x) - \frac{1}{3} \sin^3(x) + C}$

Anti-derivative of $\cos^5(x)$

1. Start with $\cos^4(x) \cos(x)$.
2. Use $\cos^4(x) = (\cos^2(x))^2$ and $\cos^2(x) = 1 - \sin^2(x)$.
3. Substitute $u = \sin(x)$, $du = \cos(x) \ dx$.
4. Integral becomes $\int (1 - u^2)^2 du$. Expand and integrate: $\int (1 - 2u^2 + u^4) du$.
5. Anti-derivatives are $u - \frac{2u^3}{3} + \frac{u^5}{5} + C$
6. Substitute back $u = \sin(x)$ for the final answer:
   • $\boxed{\sin(x) - \frac{2}{3} \sin^3(x) + \frac{1}{5} \sin^5(x) + C}$

Anti-derivative of $\cos^5(x) \sin(x) dx$

1. Use $u = \cos(x)$, $du = -\sin(x) \ dx$.
2. Integral becomes $- \int u^5 du$.
3. Anti-derivative: $- \frac{u^6}{6} + C$
4. Substitute back $u = \cos(x)$ for the final answer:
   • $\boxed{- \frac{1}{6} \cos^6(x) + C}$

Anti-derivative of $\sin^5(x) \cos^2(x)$

1. Use $u = \cos(x)$, $du = -\sin(x) \ dx$.
2. Rewrite integral in terms of $u$:
   • $ \sin^5(x) = (1 - \cos^2(x))^2 \sin(x) \cos^2(x)$.
   • Substitute: $(1 - u^2)^{2} u^2 (-du)$
3. Expand and integrate:
   • $- \int (u^2 - 2u^4 + u^6) du $.
4. Anti-derivatives: $- \left( \frac{u^3}{3} - \frac{2u^5}{5} + \frac{u^7}{7} \right) + C$
5. Substitute back $u = \cos(x)$:
   • $ \boxed{- \frac{1}{3} \cos^3(x) + \frac{2}{5} \cos^5(x) - \frac{1}{7} \cos^7(x) + C}$

Anti-derivative of $\sin^2(x)$

1. Use power reducing formula: $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$
2. Integral becomes $\frac{1}{2} \int (1 - \cos(2x)) dx$.
3. Anti-derivatives are $\frac{1}{2} (x - \frac{\sin(2x)}{2}) + C$
4. Simplify to:
   • $\boxed{\frac{x}{2} - \frac{\sin(2x)}{4} + C}$

Anti-derivative of $\cos^2(3x)$

1. Use power reducing formula: $\cos^2(3x) = \frac{1}{2}(1 + \cos(6x))$
2. Integral becomes $\frac{1}{2} \int (1 + \cos(6x)) dx$
3. Anti-derivatives: $\frac{1}{2} \left( x + \frac{\sin(6x)}{6} \right) + C$
4. Simplify to:
   • $\boxed{\frac{x}{2} + \frac{\sin(6x)}{12} + C}$

Anti-derivative of $\sin^4(x)$

1. Use power reducing formula twice: $\sin^4(x) = (\sin^2(x))^2 = \left(\frac{1}{2}(1 - \cos(2x))\right)^2$
2. Integral involves expanding: $\frac{1}{4}(1 - \cos^2(2x)) = \frac{1}{4}(1 - \frac{1}{2}(1 + \cos(4x)))$
3. Final integral to solve is: $\frac{1}{4} \int \left( \frac{3}{2} - \frac{1}{2} \cos(4x) \right) dx$
4. Anti-derivatives:
   • $\boxed{\frac{3x}{8} - \frac{\sin(4x)}{32} + C}$

Anti-derivative of $\sin^2(x) \cos^2(x)$

1. Use $\sin^2(x) \cos^2(x) = \left(\frac{1}{2}(1 - \cos(2x))\right)\\ imes\\left(\frac{1}{2}(1 + \cos(2x))\right)$
2. Rewrite as: $\frac{1}{4} \int (1 - \cos^2(2x))dx$
3. Use power reducing again:
   • $1 - \cos^2(2x) = \sin^2(2x) = \frac{1}{2} \left(1 - \cos(4x)\right)$
4. Final integral to solve: $\frac{1}{8} \int (1 - \cos(4x)) dx$
5. Anti-derivatives: $\frac{1}{8}(x - \frac{\sin(4x)}{4}) + C$
6. Simplify to:
   • $\boxed{\frac{x}{8} - \frac{\sin(4x)}{32} + C}$

Anti-derivative of $\cos^2(x)\tan^3(x)$

1. Convert $\tan(x)$ into $\frac{\sin(x)}{\cos(x)}$:
   • $ \int \cos^2(x)\frac{\sin^3(x)}{\cos^3(x)}$
2. Simplify to: $ \int \frac{\sin^3(x)}{\cos(x)}$
3. Use $u = \cos(x)$, $du = -\sin(x) dx$
4. Integral becomes: $ \int \frac{(1-u^2)}{u} (-du)$
5. Simplify and integrate: $- \left( \int \frac{1}{u} du - \int u du \right)$
6. Result: $-\left( \ln|u| - \frac{u^2}{2} \right) + C$
7. Substitute $u = \cos(x)$:
   • $ \boxed{-\ln|\cos(x)| + \frac{\cos^2(x)}{2} + C}$