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If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can always be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions of a complex variable.
In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of ) are those with this form. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of ...
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions).
A general solution containing the arbitrary constant is often necessary to identify the correct particular solution. For example, to obtain the antiderivative of cos ( x ) {\displaystyle \cos(x)} that has the value 400 at x = π, then only one value of C {\displaystyle C} will work (in this case C = 400 {\displaystyle C=400} ).
When evaluating definite integrals by substitution, one may calculate the antiderivative fully first, then apply the boundary conditions. In that case, there is no need to transform the boundary terms. Alternatively, one may fully evaluate the indefinite integral first then apply the boundary conditions. This becomes especially handy when ...
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