u sub

renegade05

Full Member
Joined
Sep 10, 2010
Messages
260
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.
 
renegade05 said:
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 x=t2,   dx=2tdt\displaystyle x=t^{2}, \;\ dx=2tdt

Then it becomes:

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

Now, use parts:

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

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

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

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

2xcos(x)+2sin(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.
 
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