Title: Indefinite Integral - Examples of specific types of problems

• Introduction to indefinite integrals
• Importance of finding antiderivatives
• Overview of specific types of problems covered in this lecture

Type 1: Basic Power Rule

• Formula: $\int x^n , dx = \frac{{x^{n+1}}}{{n+1}} + C$
• Examples:
  • $\int x^3 , dx = \frac{{x^4}}{4} + C$
  • $\int x^{-2} , dx = -\frac{1}{{x}} + C$

Type 2: Exponential Functions

• Formula: $\int e^x , dx = e^x + C$
• Examples:
  • $\int e^{2x} , dx = \frac{{e^{2x}}}{2} + C$
  • $\int e^{-3x} , dx = -\frac{1}{{3}} e^{-3x} + C$

Type 3: Trigonometric Functions

• Formula: $\int \sin(x) , dx = -\cos(x) + C$
• Examples:
  • $\int \sin(2x) , dx = -\frac{{\cos(2x)}}{2} + C$
  • $\int \cos(3x) , dx = \frac{{\sin(3x)}}{3} + C$

Type 4: Logarithmic Functions

• Formula: $\int \frac{1}{x} , dx = \ln|x| + C$
• Examples:
  • $\int \frac{1}{2x} , dx = \ln|2x| + C$
  • $\int \frac{1}{x+1} , dx = \ln|x+1| + C$

Type 5: Rational Functions

• Formula: Integration by parts or partial fraction decomposition may be required
• Examples:
  • $\int \frac{1}{x^2 + 1} , dx$ (partial fraction decomposition needed)
  • $\int x \ln(x) , dx$ (integration by parts needed)

Type 6: Trigonometric Substitution

• Formula: Used for integrating expressions containing square roots and trigonometric functions
• Substitutions:
  • $x = \sin(t)$
  • $x = \cos(t)$
  • $x = \tan(t)$
• Examples:
  • $\int \frac{{dx}}{{\sqrt{1-x^2}}}$
  • $\int \sqrt{1+x^2} , dx$

Type 7: Hyperbolic Functions

• Formula: $\int \sinh(x) , dx = \cosh(x) + C$
• Examples:
  • $\int \sinh(2x) , dx = \frac{{\cosh(2x)}}{2} + C$
  • $\int \cosh(3x) , dx = \frac{{\sinh(3x)}}{3} + C$

Type 8: Integration by Parts

• Formula: $\int u , dv = uv - \int v , du$
• Examples:
  • $\int x \cos(x) , dx$
  • $\int x^2 e^x , dx$

Type 9: Improper Integrals

• Formula: Evaluate limits as the interval approaches infinity or a point of discontinuity
• Examples:
  • $\int_0^\infty e^{-x} , dx$
  • $\int_{-\infty}^{\infty} \frac{1}{1+x^2} , dx$

Type 10: Integration by Substitution

• Formula: $\int f(g(x))g’(x) , dx = \int f(u) , du$, where $u=g(x)$
• Examples:
  • $\int 2x \cos(x^2) , dx$
  • $\int e^{\sqrt{x}} , dx$

Type 11: Integration of Composite Functions

• Formula: $\int f(g(x)) , dx = F(g(x)) , dx$
• Examples:
  • $\int \sqrt{1-\sin(x)} , dx$
  • $\int \frac{1}{\sqrt{1-x^2}} , dx$

Type 12: Integration of Rational Functions

• Formula: Use partial fraction decomposition to simplify the expression
• Examples:
  • $\int \frac{2x+3}{x^2-4x+4} , dx$
  • $\int \frac{x^3-1}{x-1} , dx$

Type 13: Integration of Trigonometric Products

• Formula: Use product-to-sum identities or trigonometric identities to simplify the expression
• Examples:
  • $\int \sin^2(x) \cos^3(x) , dx$
  • $\int \tan(x) \sec^2(x) , dx$

Type 14: Integration of Exponential Functions

• Formula: Apply appropriate substitution or integration by parts
• Examples:
  • $\int e^x \cos(x) , dx$
  • $\int e^{2x} \sin(3x) , dx$

Type 15: Integration of Logarithmic Functions

• Formula: Integration by parts or substitution may be required
• Examples:
  • $\int x \ln(x) , dx$
  • $\int \sin^{-1}(x) , dx$

Type 16: Integration of Hyperbolic Functions

• Formula: Use appropriate hyperbolic identities or integration techniques
• Examples:
  • $\int \sinh^3(x) , dx$
  • $\int \cosh(2x) \sinh(2x) , dx$

Type 17: Integration of Trigonometric Powers

• Formula: Use various trigonometric identities and manipulation techniques
• Examples:
  • $\int \sin^3(x) \cos^2(x) , dx$
  • $\int \tan^2(x) \sec^3(x) , dx$

Type 18: Inverse Trigonometric Functions

• Formula: Use appropriate substitution or trigonometric identities
• Examples:
  • $\int \frac{1}{\sqrt{1-x^2}} , dx$
  • $\int \frac{1}{1+x^2} , dx$

Type 19: Integration of Powers of Trigonometric Functions

• Formula: Simplify using power reduction formulas or appropriate manipulation
• Examples:
  • $\int \sin^5(x) , dx$
  • $\int \cos^4(x) , dx$

Type 20: Integration of Inverse Trigonometric Functions

• Formula: Use appropriate substitution or trigonometric identities
• Examples:
  • $\int \frac{1}{\sqrt{1-x^2}} , dx$
  • $\int \frac{1}{1+x^2} , dx$

Type 21: Integration of Powers of Trigonometric Functions

• Formula: Simplify using power reduction formulas or appropriate manipulation
• Examples:
  • $\int \sin^5(x) , dx$
  • $\int \cos^4(x) , dx$

Type 22: Integration of Rational Expressions with Trigonometric Functions

• Formula: Use trigonometric identities or appropriate substitution
• Examples:
  • $\int \frac{\sin^2(x)}{\cos^4(x)} , dx$
  • $\int \frac{\sec(x)}{\tan^2(x)} , dx$

Type 23: Integration involving Absolute Values

• Formula: Split into cases and solve each separately
• Examples:
  • $\int |x-2| , dx$
  • $\int |x^2-4| , dx$

Type 24: Integration of Hyperbolic Trigonometric Functions

• Formula: Use corresponding hyperbolic identities or appropriate techniques
• Examples:
  • $\int \sinh^2(x) , dx$
  • $\int \cosh(2x) \sinh(3x) , dx$

Type 25: Integration of Logarithmic-exponential Functions

• Formula: Apply integration by parts or substitution
• Examples:
  • $\int x \ln(e^x) , dx$
  • $\int e^x \ln(x) , dx$

Type 26: Integration of Trigonometric Functions raised to a Power

• Formula: Use reduction formula or appropriate manipulation techniques
• Examples:
  • $\int \sin^3(x) \cos^2(x) , dx$
  • $\int \tan^2(x) \sec^3(x) , dx$

Type 27: Integration of Hyperbolic Functions raised to a Power

• Formula: Use reduction formula or appropriate manipulation techniques
• Examples:
  • $\int \sinh^3(x) \cosh^2(x) , dx$
  • $\int \tanh^2(x) \sech^3(x) , dx$

Type 28: Integration of Arc Length

• Formula: Apply integration techniques to calculate arc length
• Examples:
  • $\int \sqrt{1+(\cos(x))^2} , dx$
  • $\int \sqrt{1+(\sinh(x))^2} , dx$

Type 29: Integration of Logarithmic Trigonometric Functions

• Formula: Use substitution or appropriate manipulation techniques
• Examples:
  • $\int \ln(\sin(x)) , dx$
  • $\int \ln(\cos(x)) , dx$