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Evaluate the integral of (2x+3) / ((x-2)(x+5))

by RoRi

Compute the following integral.

    \[ \int \frac{2x+3}{(x-2)(x+5)} \, dx. \]

First, we find the partial fraction decomposition of the integrand,

    \[ \frac{2x+3}{(x-2)(x+5)} = \frac{A}{x-2} + \frac{B}{x+5}. \]

This implies

    \[ A(x+5) + B(x-2) = 2x+3. \]

Plugging in the values x = -5 and x = 2 we obtain the equations

    \begin{align*} -7B &= -7 & \implies \quad B = 1 \\ 7A &= 7 & \implies \quad A = 1 \end{align*}

Therefore we have

    \begin{align*} \int \frac{2x+3}{(x-2)(x+5)} \, dx &= \int \frac{1}{x-2} \, dx + \int \frac{1}{x+5} \, dx \\[10pt] &= \log |x-2| + \log |x+5| + C. \end{align*}

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