Integration

Basic Concepts of Integration

  • Integrals : Mathematical tools used to find the accumulated total or area under curves. They can be described as the “reverse operation” of differentiation.
  • Indefinite Integral : An integral that does not have definite limits of integration. This often results in the general solution plus a constant of integration.
  • Definite Integral : An integral with specific limits of integration, which gives a numerical answer. It is often interpreted as the area under a curve.
  • Integral sign (∫) : The symbol used to denote integration. It’s a stretched ‘S’ shape, chosen to stand for ‘summation’.
  • Constant of Integration (C) : A constant added to solution of indefinite integrals due to the reversal of differentiation and to cover all solutions.
  • Fundamental Theorem of Calculus : A theorem that establishes a connection between integrals and derivatives, showing that they are reverse processes.

Techniques for Integration

  • Power Rule for Integration : This rule allows us to integrate powers of x, stating if n≠-1 then ∫x^n dx = x^(n+1)/(n+1) + C.
  • Integration by Substitution : A technique for transforming complicated integrals into simpler ones. Involves substituting a part of the function with a new variable.
  • Integration by Parts : A method used for integrating products of functions. It is essentially the reverse process of the product rule for differentiation.
  • Partial Fraction Decomposition : This technique involves breaking down a complex fraction into simpler fractions to make it easier to integrate.

Integrating Trigonometric Functions

  • Integration of Sine and Cosine : The integral of sin(x) dx is -cos(x) + C and the integral of cos(x) dx is sin(x) + C.
  • Integration of Tangent and Cotangent : The integral of tan(x) is -ln cos(x) + C and of cot(x) is ln sin(x) + C.
  • Integration of Secant and Cosecant : These follow more complex rules. For instance, the integral of sec(x) dx is ln sec(x)+tan(x) + C.

Applications of Integration

  • Area Under a Curve : Definite integrals can be used to compute the exact area under the curve of a function and above the x-axis.
  • Volume of Revolution : Using methods such as disk/washer and cylindrical shell, integration can calculate the volumes of solid bodies.
  • Solving Differential Equations : Integration is used to find the general solution of differential equations.
  • Kinematics : Integrate acceleration to find velocity, and velocity to find displacement in physics problems.

Special Integrals

  • Improper Integrals : Integrals where the function is not defined at an end point of the interval or the interval is infinite.
  • Integration of Exponential Functions : The integral of e^x dx is e^x + C.
  • Integration of Logarithmic Functions : The antiderivative of ln(x) is x(ln(x) - 1) + C.