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 2 with an irreducible polynomial function. So no need to perform the long-division. One may also use the table for this integration.

We have

\begin{displaymath}\frac{x+1}{x^2 + 1} = \frac{x}{x^2 + 1} + \frac{1}{x^2 + 1}\end{displaymath}

which implies

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


\begin{displaymath}\int \frac{1}{x^2 + 1}dx = \arctan(x)\end{displaymath}


\begin{displaymath}\int \frac{x}{x^2 + 1}dx = \frac{1}{2}\int \frac{2x}{x^2 + 1}dx = \frac{1}{2} \ln(x^2+1),\end{displaymath}

we get

\begin{displaymath}\int \frac{x+1}{x^2 + 1}dx = \frac{1}{2} \ln(x^2+1) + \arctan(x) + C\;.\end{displaymath}

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

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