Integration approaches

**fatlucky** · January 9th 2011, 07:17 AM

Hey,

What is the best approach to integrate this:

sqrt[ (t+a)^2 + a^2 ] dt

Thanks in advance

**alexmahone** · January 9th 2011, 07:28 AM

Originally Posted by fatlucky

Hey,

What is the best approach to integrate this:

sqrt[ (t+a)^2 + a^2 ] dt

Thanks in advance

Integrate by parts.

$u=\sqrt{(t+a)^2+a^2}$ and $dv=1$

**Random Variable** · January 9th 2011, 07:32 AM

First let $u = t+a$ . The let $u = a \tan x$

**TheCoffeeMachine** · January 9th 2011, 07:42 AM

Put $(t+a) = a\sinh{\varphi}$ .

**fatlucky** · January 9th 2011, 07:55 AM

Damn.. I was thinking there would be a more obvious substitution I could use..
I would not think to use these in a calculus exam :O
Thanks for the suggestions - I'll be trying them out now and see what works best for me

**chisigma** · January 9th 2011, 08:15 AM

In my opinion the best way is the simple substution $x= \frac{t}{a}$ so that the integral becomes...

$\displaystyle a^{2}\ \int \sqrt{1+ x^{2}}\ dx$ (1)

... that can be solved by parts in some steps...

Kind regards

$\chi$ $\sigma$

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