How do you integrate (x+1)/(x^2+2)?

Question

I know the answer is 0.5log(x^2+2)+(arctan(x/root(2)))/root(2… but I don't know how to get it.
Thanks

Answer 1

ʃ(x + 1) / (x² + 2) dx

Apply integral sum rule

ʃ x / ( x² + 2) dx + ʃ1 / ( x² + 2) dx

put x² + 2 = t

Apply derivative to each side

2xdx = dt

xdx = (1 / 2)dt -(1)

x² = t - 2

Take square root  to each side

x = √(t - 2)

dx = dt / 2√(t - 2) -(2)

Substitute xdx = (1 / 2)dt and dx = dt / 2√(t - 2) in the above integral

ʃ(1 / 2)dt / t + ʃdt / (2√(t - 2))(1 / t)

1 / 2 log|t| + C + ʃdt / (2t√(t - 2))

Substitute t = x² + 2

1 / 2 log|x² + 2| + C + ʃdx / (x ² + (√2)²)

1 / 2 log|x² + 2| + C + (1 / √2) tan^-1(x /√2).