# How do I find the integral #int(x-9)/((x+5)(x-2))dx# ?

This integral can be solved by using the Partial Fractions approach, giving an answer of

Process:

The partial fractions approach is useful for integrals which have a denominator that can be factored but not able to be solved by other methods, such as Substitution. This equation already has its denominator factored, but note that if we were instead given the multiplied form:

we would need to factor the denominator to continue. We can now turn this function into its partial fraction equivalent:

Using these values in our original partial fractions representation from above, we have:

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To find the integral of ( \frac{x-9}{(x+5)(x-2)} ) with respect to ( x ), you can use partial fraction decomposition. First, you need to decompose the rational function into partial fractions. Then, you integrate each term separately.

The partial fraction decomposition of ( \frac{x-9}{(x+5)(x-2)} ) is:

( \frac{x-9}{(x+5)(x-2)} = \frac{A}{x+5} + \frac{B}{x-2} )

To find ( A ) and ( B ), multiply both sides by ( (x+5)(x-2) ) to clear the fractions, then equate coefficients of like terms.

Solving for ( A ) and ( B ), you'll find:

( A = -\frac{2}{7} ) and ( B = \frac{5}{7} )

Now, you integrate each term separately:

( \int \frac{-2}{x+5} ,dx + \int \frac{5}{x-2} ,dx )

This yields:

( -2\ln|x+5| + 5\ln|x-2| + C )

Where ( C ) is the constant of integration.

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