# Trig sine substitution doesnt work

#### stunner5000pt

how would on integrate $$\int_{0}^{1} \sqrt{1 + 4t^2} dt$$
trig sub sittution doesnt work since one doesnt get tan^2 +1 .

i tryed solving this with matematica and it yielded something with a sinh argument. I am not familiarwi the hyp sine substitution.

#### George Jones

Staff Emeritus
Gold Member
Use the sustitution $2t = tan \theta$. Be prepared to integrate $sec^3 \theta$.

Regards,
George

#### CarlB

Homework Helper
You may find that a substitution of t for sinh(theta)/2 will assist. And then you might find it useful to know how to write cosh(theta) in terms of exponentials.

Lots of ways to skin a small furry critter.

Carl

#### GCT

Homework Helper
to make things appear simpler factor out the 4, then make the proper trig substitution; if you still don't get it, you've definitely got some more studying to do.

#### George Jones

Staff Emeritus
Gold Member
CarlB said:
Lots of ways to skin a small furry critter.
Carl
AbsolutelY!

Another way is to make the substitution $2t = sinh x$, and then to make a "half-angle" substitution for $cosh^{2} x$.

Regards,
George

#### stunner5000pt

would the limits of integration change in this process?
for t=0, theta = 0
for t =1 , theta = arctan 2
is that right?

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