1. ## Integrating By Parts

My brain has apparently turned to mush since last summer.

$\displaystyle \int{(3t+5)cos(t/4)}dt$

I took...

$\displaystyle u=3t+5$
and...
$\displaystyle dv=cos(t/4)$

I need help going from there to...

$\displaystyle du=$
and
$\displaystyle v=$

the tutorial I'm using skips the intervening steps and just gives me v and du. I want to understand the underlying process. Acctual steps to get to du and v would be greatly appreciated. Thanks anyone and everyone

2. Perhaps the best way is to compute separately...

$\displaystyle \displaystyle \int (3t + 5)\ \cos \frac{t}{4}\ dt = \int 5 \ \cos \frac{t}{4}\ dt + \int 3 t\ \cos \frac{t}{4}\ dt$ (1)

The first integral doesn't require integration by part and its result is useful for the second integral...

Kind regards

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

3. To get du, differentiate the expression for u. Similarly, to get v, integrate the expression for dv.
Note that as chisigma says, there is no reason to choose 3t + 5 as u, since the integral can be simplified first.