Problems on Techniques of Integration

We have

\begin{displaymath}\frac{3x-2}{x^2 + 1} = \frac{3x}{x^2 + 1} - \frac{2}{x^2 + 1}\end{displaymath}

which implies

\begin{displaymath}\int \frac{3x-2}{x^2 + 1}dx = \int \frac{3x}{x^2 + 1}dx - \int \frac{2}{x^2 + 1}dx\cdot\end{displaymath}

Since

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

and

\begin{displaymath}\int \frac{3x}{x^2 + 1}dx = \frac{3}{2} \ln(x^2+1),\end{displaymath}

we get

\begin{displaymath}\int \frac{3x-1}{x^2 + 1}dx = \frac{3}{2} \ln(x^2+1) -2 \arctan(x) + C\;.\end{displaymath}

Detailed Answer.


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