Question

how to find this integral?

Answer 1

\[\int\limits_{}^{} \frac{ x^{2} }{ \sqrt{x^{2} + c^{2}} } dx\], c is the constant

Answer 2

i would use trigonometric substitution ... i would check the property of tan or cot

Answer 3

ok let me figure it out a while

Answer 4

if i use \[\frac{ x }{ d } = \cot \], i end up getting this
\[-c \int\limits_{}^{} \cot^{2} \theta \csc^{2} \theta d \theta \]

then u use substitution u = csc theta

then i get, and current stuck at here:
\[c \int\limits_{}^{} \cos \theta u du\]

am i in the right path?

Answer 5

hmm ...
x = c cot(theta) or c tan(theta) ... use tan because it's nicer

Answer 6

ok

Answer 7

my current answer is
\[\frac{ x \sqrt{x^{2}+d^{2}} }{ 2 } + \frac{i d^{2} Arcsin \left( \frac{ \sqrt{x^{2}+d^{2}} }{ d } \right) }{ 2 }\]

the first part of my answer looks like the wolframalpha's answer, but the second part...i don't know what is happening... http://www.wolframalpha.com/input/?i=integral+x2+%2F+sqrt%28x2+%2B+9%29

Answer 8

show me your steps
the answer is
\[ \frac{1}{2} x \sqrt{c^2+x^2}-\frac{1}{2} c^2 \log \left(\sqrt{c^2+x^2}+x\right) \]

Answer 9

got it, thanks