Indefinite Integral Indefinite Integral The indefinite integral is defined as a function that will describe an area under the function’s curve from an undefined point to another arbitrary point. The absence of a specified first point leads to an arbitrary constant most often denoted as C and is always considered a part of indefinite integral. In other words, the indefinite integral is the family of all functions whose derivative is f(x) but with a possibly finite set of exceptions. \frac{d}{dx} \int f(x) dx = f(x) Most often the theorem applies to only bound functions. But, we have to remember that if ‘f’ is continuous on (a, b) then it is uniform, continuous and thus bounded on any sub- interval [c, d] \subset (a, b). We will not get an indefinite integral that is Lipschitz on all of (a, b) unless ‘f’ is bounded. Theorem: Suppose f(a, b) \to R is a function on an open interval (a, b) which could be bounded or un-bounded and that there are no discontinuity points of ‘f’ in (a, b) then ‘f’ has an indefinite integral on (a, b).
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Indefinite integrals are integrals without limits. The technique of integration is very useful in two ways. To find the function whose derivative is given. To find the area bounded by a curve given by the function under certain conditions. Let f(x) be the derivative of the function F(x) with respect to x on an interval. Then, the collection of all antiderivatives of f(x) is called indefinite integral of f(x) with respect to x and is denoted by, \int f(x) dx = F(x) + c Here, c is called the constant of integration. Study Tips to Indefinite Integrals The symbol ∫ is an elongated form of S and represents a summation. ∫(Symbol for integration) f(x)(Integrand) dx(Differential of x) = F(x)(Antiderivative or indefinite integral of f(x)) + C(Constant of integration). The integral of a function is not unique. They differ by a constant. Integration is the inverse of differentiation. Formula for Indefinite IntegralsBack to Top Given below are some Indefinite Integral Formulas. Indefinite Integral Formula for Exponential Functions ∫exdx=ex ∫eaxdx=1aeax ∫eaxcosbxdx=eaxa2+b2(acosbx+bsinbx) ∫eaxsinbxdx=eaxa2+b2(asinbx−bcosbx) ∫axdx=axlna+c The above formulas are helpful to learn the integral of an exponential function.
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Indefinite Integral Formula for Logarithmic Functions ∫logxdx=x(logx−1)+c ∫1xdx=log|x|+c ∫axdx=axlogx+c ∫dxa2−x2=12aloga+xa−x+c ∫dxx2−a2=12alogx−ax+a+c The above formulas are helpful to learn the integral of logarithmic function.
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