u sub

[renegade05]

Is it possible to do these ones with u substitution? or do you need to integrate by parts?

\(\displaystyle \[ \int \sin \sqrt{\theta} \,d\theta\)

and

\(\displaystyle \[ \int \frac{5x}{\sqrt[3]{1-2x}} dx\)

and how the heck would you do them if so? I dont know where to start.

 

[galactus]

Is it possible to do these ones with u substitution? or do you need to integrate by parts?

\(\displaystyle \[ \int \sin \sqrt{\theta} \,d\theta\)

One way would be to use sub and parts.

Let \(\displaystyle x=t^{2}, \;\ dx=2tdt\)

Then it becomes:

\(\displaystyle 2\int tsin(t)dt\)

Now, use parts:

\(\displaystyle u=t, \;\ dv=sin(t)dt, \;\ du=dt, \;\ v=-cos(t)\)

\(\displaystyle 2(-tcos(t)+\int cos(t)dt)\)

\(\displaystyle -2tcos(t)+2sin(t)\)

Resub \(\displaystyle t=\sqrt{x}\)

\(\displaystyle 2\sqrt{x}cos(\sqrt{x})+2sin(\sqrt{x})\)


A simple u sub will work for the other one. Let u=1-2x is one way.