SOLUTION: Find integral (dx/sqr(x^2+16))

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Question 1121154: Find integral (dx/sqr(x^2+16))
Found 3 solutions by Alex.33, solver91311, Edwin McCravy:
Answer by Alex.33(110) About Me  (Show Source):
You can put this solution on YOUR website!
https://www.symbolab.com/solver/step-by-step/%5Cint%5Cleft(%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E%7B2%7D%2B16%7D%7D%5Cright)dx
check this out. This may solve your problem. If you have any question about the proofs of the specific formulae, just ask the search engines-there are abundant of them awaiting.
Special mention: it seems that your question turns out to be arcsinh(x/4) on Methematica. But it seems that the result given by the two sites are equivalent. I derived both using both tools, and got the same result.

Answer by solver91311(24713) About Me  (Show Source):
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Find integral (dx/sqr(x^2+16))
int%28dx%2Fsqrt%28x%5E2%2B16%29%29

We use trig substitution by drawing this right triangle:

 

Then x%2F4=tan%28theta%29 and sqrt%28x%5E2%2B16%29%2F4=cos%28theta%29

So

x=4tan%28theta%29 and sqrt%28x%5E2%2B16%29=4sec%28theta%29
                        
dx=4sec%5E2%28theta%29d%28theta%29

Substituting, int%28dx%2Fsqrt%28x%5E2%2B16%29%29 becomes

int%28%284sec%5E2%28theta%29d%28theta%29%29%2F4sec%28theta%29%29

Cancel and get

int%28sec%28theta%29d%28theta%29%29

We use 

int%28sec%28u%29du%29%22%22=%22%22ln%28sec%28u%29%5E%22%22%2Btan%28u%29%29%22%22%2B%22%22C

int%28sec%28theta%29d%28theta%29%29%22%22=%22%22ln%28sec%28theta%29%5E%22%22%2Btan%28theta%29%29%22%22%2B%22%22C%5B1%5D%22%22=%22%22

Use trig ratios from above right triangle:

ln%28sqrt%28x%5E2%2B16%29%2F4%2Bx%2F4%29%22%22%2B%22%22C%5B1%5D%22%22=%22%22ln%28expr%281%2F4%29%28sqrt%28x%5E2%2B16%29%2Bx%29%5E%22%22%29%22%22%2B%22%22C%5B1%5D%22%22=%22%22ln%281%2F4%29%2Bln%28sqrt%28x%5E2%2B16%29%2Bx%29%22%22%2B%22%22C%5B1%5D%22%22=%22%22

We let C%22%22=%22%22ln%281%2F4%29%2BC%5B1%5D to simplify and end up with:

ln%28sqrt%28x%5E2%2B16%29%2Bx%29%22%22%2B%22%22C

Edwin