Integration of Rational functions: Example1

Find

Solution. Since the degree of the numerator is higher than the denominator, we should perform the long-division. We get

which implies

We concentrate on the fraction . The partial decomposition technique gives

This gives x + 2 = A(x-1) + B(x+1). If we substitute x=1, we get B = 3/2 and we substitute x=-1, we get A = -1/2. Therefore, we have

Now we are in position to perform the desired integration. Indeed, we have

Since we have ,

we get

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