Question

please help me find:
the integral of sqrt(1 + 4x^2) using hyperbolic substitution

Answer 1

i think the trick is to factor the 4 out of the whole thing yes?

Answer 2

get \[\sqrt{4({\frac{1}{4}+x^2)}}\]

Answer 3

then write
\[2\sqrt{\frac{1}{4}+x^2}\]

Answer 4

and the substitution you use for
\[\sqrt{a^2+x^2}\] is
\[x=a\sinh(u)\]

Answer 5

then you do a bunch of mess about rewriting as cosh(u) and then substiting back much easier to look in a table of integrals and see that
\[\int\sqrt{a^2+x^2}dx = \frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\ln(x+\sqrt{a^2+x^2})+C\]

Answer 6

hahaa finally got it, thanks !

Answer 7

welcome. sorry i was not much help