Question

Find:

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

Answer 1

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

Using partial fraction,

`(x^2 + x + 1)/((x + 2)(x^2 + 1)) = A/(x + 2) + (Bx + C)/(x^2 + 1)`

`(x^2 + x + 1)/((x + 2)(x^2 + 1)) = (A(x^2 + 1) + (Bx + C)(x + 2))/((x + 2)(x^2 + 1))`

x² + x + 1 = Ax² + A + Bx² + 2Bx + Cx + 2C

x² + x + 1 = A(x² + 1) + (Bx + C)(x + 2)

Comparing the coefficient of x², x, and the constant.

1 = A + B

1 = 2B + C

Solving we get, 1 = A + 2C

Find Let A Let x = −2.

x² + x + 1 = A(x² + 1)

(−2)² + (−2) + 1 = A((−2)² + 1)

4 − 2 + 1 = A(4 + 1)

3 = A 5

A = `3/5`

Find B and C, equate coefficients from x² + x + 1 = (A + B)x² + (2B + C)x + (A + 2C).

Coefficients of x² : 1 = A + B 

1 = `3/5 + B`

B = `1 - 3/5`

B = `2/5`

Constant terms: 1 = A + 2C

`3/5 + 2C = 1`

`2C = 1 - 3/5`

`2C = (5 - 3)/5`

`2C = 2/5`

C = `1/5`

Hence, 

`(x^2 + x + 1)/((x + 2)(x^2 + 1)) = 3/(5(x + 2)) + (2x + 1)/(5(x^2 + 1))`

∴ I = `int(3)/(5(x 
+ 2)) dx + int(2x + 1)/(5(x^2 + 1)) dx`

`int3/5(dx)/(x + 2) + 1/5int(2x)/(x^2 + 1) dx + 1/5int(dx)/(x^2 + 1)`

x² + 1 = t

2xdx = dt

= `3/5int(dx)/(x + 2) + 1/5int(dt)/t + 1/5 int(dt)/(x^2 + 1)`

= `3/5 log|x + 2| + 1/5 log|t| + 1/5tan^-1x + C`

= `3/5 log|x + 2| + 1/5 log|(x^2 + 1)| + 1/5tan^-1x + C`