SOLUTION: integral of xarctanx

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Question 401207: integral of xarctanx
Found 2 solutions by Alan3354, richard1234:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
= (x^2+1)*arctan(x)/2 - x/2 + C

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Applying integration by parts, if we let

u+=+arctan+x --> du+=+%281%2F%28x%5E2+%2B+1%29%29+dx

dv+=+x%2Adx --> v+=+x%5E2%2F2, then



I'll rewrite int%28x%5E2%2F2%28x%5E2%2B1%29%2C+dx%29 as int%28%281%2F2%29+-+1%2F2%28x%5E2%2B1%29%2C+dx%29, which is equal to int%281%2F2%2C+dx%29+-+%281%2F2%29int%281%2F%28x%5E2%2B1%29%2C+dx%29.

Therefore the integral is equal to



= arctan%28x%29%28x%5E2%2F2%29+-+x%2F2+%2B+%281%2F2%29%28arctan%28x%29%29+%2B+C

= %281%2F2%29%28%28x%5E2%2B1%29%28arctan%28x%29%29+-+x%29+%2B+C, where C is a constant