Problems on Techniques of Integration

We have

$$\frac{3x-2}{x^2 + 1} = \frac{3x}{x^2 + 1} - \frac{2}{x^2 + 1}$$

which implies

$$\int \frac{3x-2}{x^2 + 1}dx = \int \frac{3x}{x^2 + 1}dx - \int \frac{2}{x^2 + 1}dx\cdot$$

Since

$$\int \frac{2}{x^2 + 1}dx = 2 \arctan(x)$$

and

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

we get

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

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