Evaluate the following integral (x+2) / (x2 + x)

Compute the following integral.

$\int \frac{x+2}{x^2+x} \, dx.$

We have

$\begin{align*} \int \frac{x+2}{x^2+x} \, dx &= \int \frac{x+1 + 1}{x(x+1)} \, dx \\ &= \int \frac{1}{x} \, dx + \int \frac{1}{x(x+1)} \, dx. \end{align*}$

To evaluate the second integral on the right we use partial fractions. Write,

$\frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1}.$

This gives us the equations

$A(x+1) + Bx = 1.$

Letting $x = 0$ and $x = -1$ we have

$A = 1 \qquad \text{and} \qquad B = -1.$

Therefore, we have

$\begin{align*} \int \frac{x+2}{x^2+x} \, dx &= \int \frac{1}{x} \, dx + \int \frac{1}{x(x+1)} \, dx \\ &= \log |x| + \int \frac{1}{x} \, dx - \int \frac{1}{x+1} \, dx \\ &= \log |x| + \log |x| - \log |x+1| + C \\ &= 2 \log |x| - \log|x+1| + C. \end{align*}$

Stumbling Robot

Evaluate the following integral (x+2) / (x² + x)

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