Question

Evaluate the integral. (Use C for the constant of integration. Enter your answer using function notation - use ln(x) instead of ln x.)

Answer 1

`\int\limits {(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} \, dx = (3)/(2)\ln(x^2 + 1) + 4√(3)\tan^(-1)((x)/(\sqrt 3) )+ c`

Explanation:

Given

`\int\limits {(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} \, dx`

Required

Integrate

Using partial fraction, we have:

`(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} = (Ax+B)/(x^2 + 1) + (Cx + D)/(x^2 + 3)`

Take LCM

`(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} = ((Ax+B)(x^2 + 3)+ (Cx + D)(x^2 + 1))/((x^2 + 1)(x^2 + 3))`

Cancel out the denominators

`3x^3 + 4x^2 + 9x + 4} = (Ax+B)(x^2 + 3)+ (Cx + D)(x^2 + 1)`

Open brackets

`3x^3 + 4x^2 + 9x + 4} = Ax^3+Bx^2 + 3Ax +3B+ Cx^3 + Dx^2 + Cx + D`

Collect like terms

`3x^3 + 4x^2 + 9x + 4 = Ax^3+ Cx^3+Bx^2+ Dx^2 + 3Ax+ Cx +3B + D`

Compare like terms on opposite sides

`Ax^3 + Cx^3 = 3x^3`
`A + C = 3`

`Bx^2 + Dx^2 = 4x^2`
`B + D = 4`

`3Ax + Cx = 9x`
`3A + C = 9`

`3B + D = 4`

Subtract
`B + D = 4` from
`3B + D = 4`

`3B - B + D - D = 4 - 4`

`2B + 0 = 0`

`2B = 0`

`B = 0`

`B + D = 4`

`D =4 - B`

`D =4 - 0`

`D =4`

Subtract
`A + C = 3` from
`3A + C = 9`

`3A - A + C - C = 9 - 3`

`2A = 6`

`A = 3`

`A + C = 3`

`C = 3 - A`

`C = 3 - 3`

`C = 0`

So, we have:

`A = 3`
`B = 0`
`C = 0`
`D =4`

`(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} = (Ax+B)/(x^2 + 1) + (Cx + D)/(x^2 + 3)`

`(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} = (3x+0)/(x^2 + 1) + (0*x + 4)/(x^2 + 3)`

`(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} = (3x)/(x^2 + 1) + (4)/(x^2 + 3)`

The integral becomes:

`\int\limits {[(3x)/(x^2 + 1) + (4)/(x^2 + 3)]} \, dx`

Split:

`\int\limits {(3x)/(x^2 + 1) \, dx + \int\limits {(4)/(x^2 + 3)} \, dx`

Split

`(3)/(2) \int\limits {(2x)/(x^2 + 1) \, dx + 4\int\limits {(1)/(x^2 + 3)} \ dx`

Integrate

`(3)/(2)\ln(x^2 + 1) + 4√(3)\tan^(-1)(x)/(\sqrt 3) + c`

Hence:

`\int\limits {(3x^3 + 4x^2 + 9x + 4)/((x^2 + 1)(x^2 +3))} \, dx = (3)/(2)\ln(x^2 + 1) + 4√(3)\tan^(-1)((x)/(\sqrt 3) )+ c`