How do you integrate #int (x^3 +x^2+2x+1) / [(x^2+1) (x^2+2)]# using partial fractions?
1 Answer
Explanation:
Note: If the denominator of the partial fraction is not factorable, always remember the degree of the numerator is one less. Like the problem
Step 1: Find the equivalent partial fraction
Step 2 Solve the partial fraction by multiply by Least common denominator to get
Step 3: Multiply/expand/foil the right hand side to get
Step 4: Set up the system of equation, using the corresponding coefficient for the corresponding terms
Step 5: Solve the system of equation
#+(2 = 2A + C)#
#0 = 3C => C= 0# ,#A= 1#
#+ (B +D = 1)#
#-B= 0=> B= 0, D= 1#
Step 6: Write the equivalent partial fraction for the integral like this
Step 7 Integrate the integral
the first integral can be done by u substitution like this
let
Note:
The second integral is
Answer:
#1/2 ln (x^2 +1) +1/sqrt(2) arctan(x/sqrt2) +"Constant"#
I let my constant value to be
this is another answer: