# Integration approaches

• Jan 9th 2011, 07:17 AM
fatlucky
Integration approaches
Hey,

What is the best approach to integrate this:

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

• Jan 9th 2011, 07:28 AM
alexmahone
Quote:

Originally Posted by fatlucky
Hey,

What is the best approach to integrate this:

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

Integrate by parts.

$\displaystyle u=\sqrt{(t+a)^2+a^2}$ and $\displaystyle dv=1$
• Jan 9th 2011, 07:32 AM
Random Variable
First let $\displaystyle u = t+a$. The let $\displaystyle u = a \tan x$
• Jan 9th 2011, 07:42 AM
TheCoffeeMachine
Put $\displaystyle (t+a) = a\sinh{\varphi}$.
• Jan 9th 2011, 07:55 AM
fatlucky
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 :D
• Jan 9th 2011, 08:15 AM
chisigma
In my opinion the best way is the simple substution $\displaystyle x= \frac{t}{a}$ so that the integral becomes...

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$