Problems on Techniques of Integration

We recognize here a rational function. The technique of integrationg such functions is the partial decomposition technique. In this case, the degree of the denominator is 1 (that is a linear term).

We have the known formula

\begin{displaymath}\int \frac{u'}{u} dx = \ln\vert u\vert + C,\end{displaymath}

and since

\begin{displaymath}\int \frac{1}{2x + 1}dx = \frac{1}{2} \int \frac{2}{2x + 1}dx,\end{displaymath}

we get

\begin{displaymath}\int \frac{1}{2x + 1}dx = \frac{1}{2}\ln\vert 2x+1\vert+ C\;.\end{displaymath}

It is a common mistake to forget the constant $C$.

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