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 numerator is bigger than the degree of the denominator, therefore we must perform long-division. We have

$$\frac{3x^2 - 2x+1}{2x - 1} = \frac{3}{2} x - \frac{1}{4}+ \frac{3}{4} \frac{1}{(2x - 1)} \cdot$$

Using the known formula

$$\int \frac{u'}{u} dx = \ln\vert u\vert + C,$$

we get

$$\int \frac{1}{(2x - 1)}dx = \frac{1}{2} \int \frac{2}{(2x - 1)}dx = \frac{1}{2}\ln\vert 2x-1\vert.$$

So

$$\int \frac{3x^2 - 2x+1}{2x - 1}dx = \frac{3}{4} x^2 - \frac{1}{4}x+ \frac{3}{8}\ln\vert 2x-1\vert+ C\;.$$

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

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