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Antiderivates. Indefinite integrals.

Antiderivatives

In calculus, an antiderivative or indefinite integral, G, of a function g, is the function that can be differentiated to obtain the original function, that is, G’ = f.

G(x) = $\int g(x)dx$

Examples

$\int u^{5}\frac{du}{4} = \frac{u^{6}}{24} + C = \frac{(x^{4}+2)^{6}}{24} + C.$

$\int u^{-1}du = ln|u| + C = ln|lnx| + C.$

$\frac{d}{dx}(1+x^{2})^{1/2} = \frac{1}{2}(1+x^{2})^{-1/2}(2x)$

Uniqueness of Antiderivatives

Let F and G tow different antiderivatives of f(x). If F’ = G’, then F(x) = G(x) + C. They are unique up to a constant.

Proof. If F’ = G’ ⇒ (F-G)’ = F’ - G’ = 0 ⇒ F-G is constant ⇒ F(x) = G(x) + C.

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