Indefinite Integral Problems Lecture

Basics of Indefinite Integrals

• Integral of a Constant:

  • Integral of 4 dx is 4x + c.
  • The constant c is added because the derivative of any constant is zero.

• Integral of Pi:

  • For pi dy, the integral is pi * y + c.

• Integral of Euler's Number (e):

  • For e dz, the integral is E * Z + c.

Integration of Variables Raised to Constants

• Formula:

  • Integral of x^n is x^(n+1)/(n+1) + c.

• Examples:

  • Integral of x^2 dx is x^3/3 + c.
  • Integral of 8x^3 simplifies to 2x^4 + c.
  • Integral of 5x^6 is found using the same technique.

Integrating Polynomial Functions

• Example:
  • For x^2 - 5x + 6, integrate each term separately.
    • x^2 becomes x^3/3.
    • -5x becomes -5x^2/2.
    • 6 becomes 6x.

Special Functions

• Square Root Function:

  • Integral of sqrt(x) is rewritten as x^(1/2).
  • Result: 2/3 * x^(3/2) + c.

• Cube Root Function:

  • Integral of cube root(x^4) is 3/7 * x^(7/3) + c.

Integration Techniques

Foiling

• Example:
  • Foil and integrate 3x - 1^2 to 9x^2 - 6x + 1.
  • Resulting integral: 3x^3/3 - 3x^2/2 + x + c.

Fractions

• Example:

  • x^4 + 6x^3 / x can be separated and simplified before integrating.
  • Resulting integral: 1/4 * x^4 + 2 * x^3 + c.

• Inverse Powers:

  • Integral of 1/x^2 becomes -1/x + c.
  • Integral of 1/x is ln|x| + c.

Exponential Functions

• Form:
  • Integral of e^(kx) is e^(kx)/k + c.
  • Examples include e^4x and 8e^2x.

Trigonometric Functions

• Cosine and Sine:

  • Integral of cos(x) is sin(x) + c.
  • Integral of sin(x) is -cos(x) + c.

• Secant and Tangent:

  • Integral of sec^2(x) is tan(x) + c.
  • Integral of sec(x)tan(x) is sec(x) + c.

Substitution Methods

U-Substitution

• Example Problems:
  • For x^2 sin(x^3) dx, let u = x^3.
  • Solve using du substitution.

Trigonometric Substitution

• Example:
  • x = tan(theta) or x = sin(theta) used to simplify integrals.

Integration by Parts

• Formula:

  • ∫u dv = uv - ∫v du.

• Example Problems:

  • x e^4x uses parts with u = x and dv = e^4x dx.

Advanced Techniques

• Inverse Trig Functions:

  • Use when integrating expressions like 4/(1 + x^2) leading to 4 tan^(-1)(x) + c.

• Conclusion:

  • The lecture covered various techniques of finding indefinite integrals with constant, polynomial, exponential, trigonometric, and other special functions, applying different methods such as substitution and integration by parts.