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 as well as the numerator is 1, therefore we must perform long-division. In this case, it is quite easy since

\begin{displaymath}\frac{x - 3}{x - 4} = 1 + \frac{1}{x - 4}\cdot\end{displaymath}

Using the known formula

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

we get

\begin{displaymath}\int \frac{x - 3}{x - 4}dx = x + \ln\vert x-4\vert+ C\;.\end{displaymath}

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

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