Differentials and integrals: Indefinite integrals
Standard antiderivatives
In addition to the previously discussed rules for calculating differentials, we also have a stock of standard antiderivatives. With direct integration we find the primitive function from this list, possibly extended with simple applications of calculation rules or an educated guess. Below is our list of standard antiderivatives and we give some examples of direct integration. \[\begin{array}{||c|cl||} \hline \mathit{function} & \mathit{antiderivative} & {}\\ \hline x^p & \dfrac{1}{p+1}x^{p+1} & \text{if }p\neq -1 \\ \dfrac{1}{x+a} & \ln(|x+a|) & {} \\ a^x & \dfrac{1}{\ln(a)}a^x & \text{for each }a>0,\;a\neq 1\\[5pt] e^x & e^x & {}\\[5pt] \sin x & -\cos x & {} \\[5pt] \cos x & \sin x & {} \\[3pt] \dfrac{1}{\cos^2 x} & \tan x \\[5pt] \sinh x & \cosh x & {} \\[5pt] \cosh x & \sinh x & {} \\[5pt] \dfrac{1}{\cosh^2 x} & \tanh x & {} \\[5pt] \dfrac{1}{\sqrt{1-x^2}} & \arcsin x & {}\\[5pt] \dfrac{1}{1+x^2} & \arctan x & {} \\[5pt] \dfrac{1}{\sqrt{1+x^2}} & \mathrm{arsinh}\; x & {}\\[5pt] \dfrac{1}{1-x^2} & \mathrm{artanh}\; x & {}\\[5pt] \hline \end{array}\]
Note that the variable \(x\) is a dummy variable which may be replaced by any other available variable
Some examples
From the table we immediately get the following integrals:
\[\int x^7\,\dd x=\frac{1}{8}x^8+c\]
By applying the sum rule of integration, we also get:
\[\int (4t^3 -t)\,\dd t=t^4-\tfrac{1}{2}t^2+c\]
For the primitive of \(\cos(2t-1)\) we can guess, for example based on the above table, and try \(\sin(2t-1)\). However, the derivative of this function is according to the chain rule for differentiation equal to \(2\cos(2t-1)\) and thus by a factor of 2 too large. So: \[\int \cos(2t-1)\,\dd t=\tfrac{1}{2}\sin(2t-1)+c\]