Question

How do you integrate `int (x^3 +x^2+2x+1) / [(x^2+1) (x^2+2)]` using partial fractions?

Answer 1

`1/2 ln (x^2 +1) +1/sqrt(2) arctan(x/sqrt2) +C`

`1/2 ln(x^2+1) +sqrt2 arctan(x/sqrt2)+C`

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

`(x^3+x^2+2x+1)/((x^2+1)(x^2+2)) = (Ax+B)/(x^2 +1) + (Cx+D)/(x^2+2)`

Step 2 Solve the partial fraction by multiply by Least common denominator to get

`x^3 + x^2 +2x+1 = (Ax+B)(x^2 +2) + (x^2 + 1)(Cx+D)`

Step 3: Multiply/expand/foil the right hand side to get

`x^3 + x^2 + 2x+1= Ax^3 + 2Ax + Bx^2 + 2B + Cx^3 + Cx + Dx^2 + D`

Step 4: Set up the system of equation, using the corresponding coefficient for the corresponding terms

`x^3 " "term" " " 1= A + C`
`x^2 " "term" " " 1= B+ D`
`x " "term" " " 2= 2A +C`
`x^0 " "term" " " 1= 2B+ D`

Step 5: Solve the system of equation
`-2(1 = A + C) => -2 = -2A + 2C`

`+(2 = 2A + C)`
`0 = 3C => C= 0` , `A= 1`

`-1(2B +D = 1) => -2B -D= -1`

`+ (B +D = 1)`
`-B= 0=> B= 0, D= 1`

Step 6: Write the equivalent partial fraction for the integral like this

`int(x^3+x^2 + 2x+1)/((x^2+1)(x^2+2)) dx` = `int x/(x^2+1) dx + int 1/(x^2+2)dx`

Step 7 Integrate the integral

the first integral can be done by u substitution like this

let `u= x^2 +1`
`du = 2xdx => (du)/2 = xdx`
Note: `int(dx)/(x) = ln|x| +C " " "or" " " log|x|+C`

The second integral is `arctan` , remember `int 1/(a^2+u^2)du =1/a arctant u/a +C`

Answer:

`1/2 ln (x^2 +1) +1/sqrt(2) arctan(x/sqrt2) +"Constant"`

I let my constant value to be `C`

this is another answer: `1/2 ln (x^2 +1) +1/sqrt(2) arctan(x/sqrt2) +C` or

`1/2 ln(x^2+1) +sqrt2 arctan(x/sqrt2)+C`