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 denominator is 2 with an irreducible polynomial function. So no need to perform the long-division. One may also use the table for this integration.

We have

$$\frac{x+1}{x^2 + 4} = \frac{x}{x^2 + 4} + \frac{1}{x^2 + 4}$$

which implies

$$\int \frac{x+1}{x^2 + 4}dx = \int \frac{x}{x^2 + 4}dx + \int \frac{1}{x^2 + 4}dx\cdot$$

Since

$$\int \frac{1}{x^2 + 4}dx = \frac{1}{2}\arctan\left(\frac{x}{2}\right)$$

and

$$\int \frac{x}{x^2 + 4}dx = \frac{1}{2}\int \frac{2x}{x^2 + 4}dx = \frac{1}{2} \ln(x^2+4),$$

we get

$$\int \frac{x+1}{x^2 + 4}dx = \frac{1}{2} \ln(x^2+4) + \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C\;.$$

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

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