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.
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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.