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Cavalieri's quadrature formula; Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule ...
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
In mathematics, the definite integral ()is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
Math Major: A Table of Integrals; O'Brien, Francis J. Jr. "500 Integrals of Elementary and Special Functions". Derived integrals of exponential, logarithmic functions and special functions. Rule-based Integration Precisely defined indefinite integration rules covering a wide class of integrands; Mathar, Richard J. (2012). "Yet another table of ...
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise ...
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, [7] with Cavalieri computing the integrals of x n up to degree n = 9 in Cavalieri's quadrature formula. [8]
An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits , in the sense that if f {\displaystyle f} is a continuous function, then whenever a n → a {\displaystyle a_{n}\rightarrow a} in the domain of f {\displaystyle f} , then the limit f ( a n ) {\displaystyle ...