# Thread: Integration via substitution method

1. ## Integration via substitution method

Hi,
Im learning to do integration using the substitution method.
My problem is:
Integrate
x/[root(1+2x)]dx
(4,0) (the 4 and 0 are at the top and bottom of the integration sign (the upper and lower bound)
using the substitution method.
i let u = 1+2x
so du = 2dx (?)
then i tried substituting back into the equation:
dx = 1/2du
so..

Any help would be appreciated.. i dont know if this is right.

2. $\displaystyle \int_0^4 \sqrt{1+2x} dx$

let $\displaystyle u = 1+2x$ then $\displaystyle du=2dx$ so $\displaystyle \frac{1}{2}du=dx$

You can change your x-limits of integration to u-limits if you would like, or you could keep them in terms of x. I prefer to keep them in terms of x but if you were to change them to u-limits, you would plug 0 =x and 4=x into the u equals equation.

$\displaystyle \int_{x=0}^{x=4}\frac{1}{2}\sqrt{u}$ $\displaystyle du$

Can you finish from here? Integrate, put u back in terms of x, then evaluate from zero to four.

PS: This is exactly what you did. U-sub can be intimidating at first, but after a couple of weeks it will seem a lot easier. Trust yourself enough to make mistakes. You were on the right track EXACTLY! I bet if you had kept going you would have definitely gotten it..because I didn't do anything different from what you did in your first posting.

3. A more direct sub. is to set $\displaystyle u^2=1+2x.$

4. Originally Posted by Krizalid
A more direct sub. is to set $\displaystyle u^2=1+2x.$
Yes, but I find when I'm first trying to explain u-sub to the students I tutor, their minds can't make that leap. I think you first have to master the basics before you can jump into "seeing" the different things you could let u equal. So by letting plain old u equal what is under the radical it's a simple variable that you can manipulate and still integrate easily...I think that letting everything equal u^2 would actually be harder than just simply integration square root of u for a beginner. Any thoughts?

Hopefully that was coherent, it's been a long day.

5. Then write it backwards as $\displaystyle u=\sqrt{1+2x},$ so I think there's no problem with that.

Before seein' these stuff the student must have a full domain of differentiation techniques.

6. Originally Posted by Krizalid
Then write it backwards as $\displaystyle u=\sqrt{1+2x},$ so I think there's no problem with that.

Before seein' these stuff the student must have a full domain of differentiation techniques.

Touche! I see your point. Thank you for the enlightening posting..perhaps I will point this out to my little calc I cherubs when we talk about u-sub.