Indefinite Integrals Lecture Notes

Overview

In this lecture, we will cover various indefinite integral problems, focusing on the anti-derivatives of constants, polynomials, and functions such as exponentials and trigonometric functions.

Key Concepts

Anti-Derivative of Constants

• Integral of a constant: To find the anti-derivative of a constant, add an x and a constant c.
  • Example:
    [ \int 4 , dx = 4x + c ]
    • Derivative check: ( \frac{d}{dx}(4x + c) = 4 )

Anti-Derivative of Other Constants

• Example with pi:
  [ \int \pi , dy = \pi y + c ]
• Example with e:
  [ \int e , dz = e z + c ]

Integrating Power Functions

• Power Rule:
  [ \int x^n , dx = \frac{x^{n+1}}{n+1} + c ]
  • Example:
    [ \int x^2 , dx = \frac{x^{3}}{3} + c ]
  • Example:
    [ \int 8x^3 , dx = 2x^4 + c ]

Integrating Polynomials

• Example:
  [ \int (x^2 - 5x + 6) , dx = \frac{x^3}{3} - \frac{5x^2}{2} + 6x + c ]

Square Root Functions

• Example:
  [ \int \sqrt{x} , dx = \frac{2}{3}x^{3/2} + c ]

Trigonometric Functions

• Integral of cosine:
  [ \int \cos(x) , dx = \sin(x) + c ]
• Integral of sine:
  [ \int \sin(x) , dx = -\cos(x) + c ]

Exponential Functions

• Integral of e to a linear function:
  [ \int e^{ax} , dx = \frac{e^{ax}}{a} + c ]
  • Example:
    [ \int e^{4x} , dx = \frac{e^{4x}}{4} + c ]

Logarithmic Functions

• Integral of 1/x:
  [ \int \frac{1}{x} , dx = \ln|x| + c ]

Integration by Parts

• Formula:
  [ \int u , dv = uv - \int v , du ]
  • Example:
    [ \int x \cos(x) , dx = x \sin(x) + \cos(x) + c ]

U-Substitution

• Example:
  [ \int x^2 \sin(x^3) , dx ]
  • Let ( u = x^3 ), then ( du = 3x^2 dx )
  • Result: ( -\frac{1}{3} \cos(x^3) + c )

Important Notes

• Always remember to add the constant c when performing integration.
• Check derivatives to verify integration results.
• Use the appropriate method based on the function regarding its type (polynomial, trigonometric, exponential, logarithmic, etc.)

Practice Problems

• Integrate various types of functions to reinforce learning:
  1. ( \int (4x^3 + 8x^2 - 9) , dx )
  2. ( \int \frac{5}{x^4} , dx )
  3. ( \int \tan(x) , dx )
  4. Find the anti-derivative of ( 4/(1 + x^2) )

Conclusion

This lecture provided a comprehensive overview of indefinite integrals, including various techniques and methods to find anti-derivatives. Practice is essential to mastery.