Question

integrate x-9/((x+5)(x-2))

Answer 1

integrals are coming fast and furious. i think this one is partial fractions right?

Answer 2

yes

Answer 3

yes definitely partial fractions because it is cooked to give nice integer answer. you want
\[\frac{x-9}{(x+5)(x-2)}=\frac{a}{x+5}+\frac{b}{x-2}\]

Answer 4

lets find a.
if x = -5 the denominator of
\[\frac{a}{x+5}\] is 0, so look at your original function, put your hand over the part that says
\[x+5\] and replace x by -5
you get
\[\frac{-5-9}{-5-2}=\frac{-14}{-7}=2\] so
\[a=2\]

Answer 5

likewise to find b, put your hand over the
\[x-2\] part and replace x by 2
you get
\[\frac{2-9}{2+5}=\frac{-7}{7}=-1\] so
\[b=-1\]

Answer 6

now your integral is
\[\int\frac{2}{x+5}dx+\int\frac{-1}{x-2}dx\] which you do in your head and get
\[2\ln(x+5)-\ln(x-2)\]

Answer 7

all the work was writing
\[\frac{x-9}{(x+5)(x-2)}=\frac{2}{x+5}+\frac{-1}{x-2}\]