How do you use partial fraction decomposition to decompose the fraction to integrate #(7)/(x^2+13x+40)#?
1 Answer
Mar 7, 2018
The integral equals
Explanation:
We wish to find factors in the denominator. The trick is to find two numbers that multiply to
#I = int 7/((x+ 5)(x+ 8))dx#
Now we can decompose in partial fractions.
#A/(x+ 5) + B/(x +8) = 7/((x +5)(x + 8))#
#A(x + 8) + B(x + 5) = 7#
#Ax + 8A + Bx + 5B = 7#
#(A + B)x + (8A + 5B) = 7#
Now we have a system of equations.
#{(A + B = 0), (8A + 5B = 7):}#
Substituting the first equation into the second we see that
#8A + 5(-A) = 7#
#3A = 7#
#A = 7/3#
Now clearly
#I = int7/(3(x + 5)) - 7/(3(x + 8)) dx#
#I= 7/3ln|x +5| - 7/3ln|x + 8| + C#
#I = 7/3ln|(x + 5)/(x +8)| + C#
Hopefully this helps!