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The problem of evaluating the definite integral F ( x ) = ∫ a x f ( u ) d u {\displaystyle F(x)=\int _{a}^{x}f(u)\,du} can be reduced to an initial value problem for an ordinary differential equation by applying the first part of the fundamental theorem of calculus .
For instance, when evaluating definite integrals using the fundamental theorem of calculus, the constant of integration can be ignored as it will always cancel with itself. However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of ...
In contrast, further Newton-Cotes methods produce increasingly differing weights, eventually leading to large positive and negative weights. This is indicative of how large degree interpolating polynomial Newton-Cotes methods fail to converge for many integrals, while Romberg integration is more stable.
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} 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.
An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms).
This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured.
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. [5] Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and; application of the residue ...