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

0 votes
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
asked Apr 12, 2013 in CALCULUS by homeworkhelp Mentor

1 Answer

0 votes

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).

 

answered Apr 13, 2013 by diane Scholar

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