Problems on Techniques of Integration

We recognize here a rational function. The technique of integrationg such functions is the partial decomposition technique. One may also use the table for this integration since the degree of the denominator is 2.

The partial decomposition technique implies

$$\frac{3x-1}{x^2 + x -2} = \frac{3x-1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x + 2}\;.$$

If we multiply both sides by $(x-1)(x+2)$, we get

$$A(x+2) + B(x-1) = 3x -1$$

which implies $A = \displaystyle \frac{2}{3}$ and $B = \displaystyle \frac{7}{3}$. So

$$\frac{3x-1}{x^2 + x -2} = \frac{1}{3} \left[\frac{2}{x-1} + \frac{7}{x + 2}\right]\;.$$

Hence

$$\int \frac{3x-1}{x^2 + x -2}dx = \frac{2}{3} \int \frac{1}{x-1}dx + \frac{7}{3}\int \frac{1}{x + 2}dx$$

which implies

$$\int \frac{x+1}{x^2 + 4}dx = \frac{2}{3} \ln\vert x-1\vert + \frac{7}{3} \ln\vert x+2\vert + C\;.$$

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