Basic Formulas for Antiderivatives

From the formulas already deduced to calculate the derivative of a function
it is possible to obtain the corresponding formulas to calculate antiderivatives.
For example, because of the definition of antiderivative, since

$\begin{equation*} \frac{d\left(\sin v\right)}{dx} = \cos v\cdot \frac{dv}{dx} \quad\text{ it follows that }\quad \int\!\cos v\,dv = \sin v + C \end{equation*}$

The following formulas for antiderivatives can be readily deduced from the corresponding formulas to calculate derivatives of functions previously justified.

$\displaystyle\int\!(dv+dw) = \int\!dv + \int\!dw$
$\displaystyle\int\!a\,dv = a \int\!dv$
$\displaystyle\int\!{dv} = v + C$
$\displaystyle\int\!v^n\,{dv} = \frac{v^{n+1}}{n+1} + C$
$\displaystyle\int\!\frac{dv}{v} = \ln |v| + C$
$\displaystyle\int\!a^v\,dv = \frac{a^v}{\ln a} + C$
$\displaystyle\int\!e^v\,dv = e^v + C$

%\item $\displaystyle\int\! \ln v\,{dv} = v\,\ln v - v + C$

$\displaystyle\int\!\sin v\,dv = -\cos v + C$
$\displaystyle\int\!\cos v\,dv = \sin v + C$
$\displaystyle\int\!\sec^2v \,dv = \tan v + C$
$\displaystyle\int\!\csc^2v \,dv = -\cot v + C$
$\displaystyle\int\!\sec v\tan v \,dv = \sec v + C$

%\item $\displaystyle\int\!\sec v \,dv = \ln \left(\sec v + \tan v\right)$

$\displaystyle\int\!\frac{dv}{\sqrt{a^2 - v^2}} = \arcsin \left(\frac{v}{a}\right) + C$
$\displaystyle\int\!\frac{dv}{\sqrt{a^2 - v^2}} = -\arccos \left(\frac{v}{a}\right) + C$
$\displaystyle\int\!\frac{dv}{a^2 - v^2} = \frac{1}{a}\,\arctan\left(\frac{v}{a}\right) + C$
$\displaystyle\int\!\sinh u\cdot du = \cosh u + C$
$\displaystyle\int\!\cosh u \cdot du = \sinh u + C$
$\displaystyle\int\!\sech^2 u\cdot du = \tanh u + C$
$\displaystyle\int\!\csch^2 u\cdot du = -\coth u + C$
$\displaystyle\int\!\tanh u\cdot \sech u\cdot du = - \sech u + C$
$\displaystyle\int\!\coth u\cdot \csch u\cdot du = \csch u + C$

These formulas and many more that can be deduced by applying the appropriate algebraic techniques, are used to calculate Definite Integrals as well as to solve equations in which the function and its successive derivatives appear.

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Basic Formulas for Antiderivatives

The basic formulas to compute antiderivatives are enlisted.