# Thread: A Nasty integral -

1. ## A Nasty integral -

Any Ideas on how to Approach this one? ...

$\displaystyle \int\sqrt{[1-sech(u)]^{2}+ [-tanh(u)sech(u)]^{2}}du$

We have a $\displaystyle sech(u)$ and its derivative $\displaystyle -tanh(u)sech(u)$ and this suggests some sort of substitution maybe, but the radical makes it a bit nasty for Me.

Any Ideas? ...Thanks Heaps,

2. Originally Posted by WalkingInMud
Any Ideas on how to Approach this one? ...

$\displaystyle \int\sqrt{[1-sech(u)]^{2}+ [-tanh(u)sech(u)]^{2}}du$

We have a $\displaystyle sech(u)$ and its derivative $\displaystyle -tanh(u)sech(u)$ and this suggests some sort of substitution maybe, but the radical makes it a bit nasty for Me.

Any Ideas? ...Thanks Heaps,
Thinking out loud here, but simplify the radical...

$\displaystyle [1-sech(u)]^{2}+ [-tanh(u)sech(u)]^{2}$

$\displaystyle 1-2sech(u)+sech^2(u)+tanh^2(u)sech^2(u)$

Use trig substitution: $\displaystyle tanh^2(u)=sech^2(u)-1$

$\displaystyle 1-2sech(u)+sech^2(u)+[sech^2(u)-1]sech^2(u)$

$\displaystyle 1-2sech(u)+sech^2(u)+sech^4(u)-sech^2(u)$

$\displaystyle 1-2sech(u)+sech^4(u)$

Now, what can we do about that?

3. Originally Posted by colby2152
[snip]Now, what can we do about that?