Special types of particular integrals

Special Types of Particular Integrals

Special Functions

  • The integral of a polynomial of degree n is a polynomial of degree n+1, plus a constant.
  • The integral of a constant is the constant times x, plus a constant.
  • The integral of the exponential function, e^x, is itself e^x, plus a constant.
  • The integral of the sine is -cosine, plus a constant, and the integral of the cosine is sine, plus a constant.
  • The integral of 1/x is the natural logarithm function, _ln x _, plus a constant.

Simple Substitution

  • When faced with more complex integrals, it might be required to use a simple substitution such as u = f(x) to rewrite the integral in a simpler form.
  • The two crucial stages in simple substitution are finding du/dx and making the correct substitution.

Integration By Parts

  • For integrals of the form ∫udv, the integration by parts formula ∫udv = uv - ∫vdu is an essential technique.
  • It is useful to remember the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) for determining u when using integration by parts.

Integrals Involving Surds

  • When faced with integrals involving surds in the denominator, it is often helpful to multiply the top and bottom of the fraction by the conjugate of the denominator.
  • Another useful approach for surds is to apply the trigonometric substitution method which involves replacing a variable with a trigonometric function.

Integrals Involving Trigonometric Functions

  • For trigonometric functions, we often use trigonometric identities, e.g., sin^2(x) = 1 - cos^2(x), or mulitple angle formulas to simplify the integral before integrating.
  • Sometimes, when integrating products of sine and cosine, it might be required to use the double-angle formulas or the half-angle formulas.

Improper Integrals

  • An improper integral refers to the integral of a function over an infinite interval or a function with an infinite discontinuity within the interval of integration.
  • We handle such cases by treating the infinity as a limit and approaching the integral accordingly.
  • If the improper integral converges, it gives a finite number. If it diverges, the function grows without bound as it is integrated over the given interval.

These techniques provide a broad foundation for tackling a variety of integrals, but practice is key to recognising which strategy to apply in different scenarios.