# Thread: help with integration

1. ## help with integration

$\int_{0}^{2} t^2sin{t^3} + 2.5dt$
So I started with integration by parts.
$\int_{0}^{2} t^2sint^3dt + \int_{0}^{2} 2.5dt$
$u = t^2$
$du = 2tdt$
$dv = sint^3dt$
$v = \frac{-1}{3t^2}cos(t^3)$
but then I realized this was gonna go on for a long time. Is there an easier way to take this integral?

2. Originally Posted by steveo0
$\int_{0}^{2} t^2sin{t^3} + 2.5dt$
So I started with integration by parts.
$\int_{0}^{2} t^2sint^3dt + \int_{0}^{2} 2.5dt$
$u = t^2$
$du = 2tdt$
$dv = sint^3dt$
$v = \frac{-1}{3t^2}cos(t^3)$
but then I realized this was gonna go on for a long time. Is there an easier way to take this integral?
If you mean $\int_0^2t^2\sin\left(t^3\right)+2.5\,dt$, then we see that we have $\int_0^2t^2\sin\left(t^3\right)\,dt+2.5\int_0^2\,d t$

To solve the first integral, make the substitution $u=t^3$
The second one is very straightforward.

Can you continue from here?

3. oh! yeah. thanks

4. Hello, steveo0!

By-parts is not necessary . . .

We are concerned with: . $\int_{0}^{2} t^2\sin(t^3)\,dt$

Let: . $u \,=\,t^3\quad\Rightarrow\quad du \,=\,3t^2\,dt \quad\Rightarrow\quad t^2\,dt \,=\,\tfrac{1}{3}du$

We have: . $\int \underbrace{\sin(t^3)}_{\downarrow}\,\underbrace{( t^2\,dt)}_{\downarrow}$
Substitute: . $\int \sin u\,\left(\tfrac{1}{3}du \right) \;=\;\tfrac{1}{3}\int \sin u\,du$ . . . Got it?