Homework Help: Substitution with integration

1. Aug 10, 2007

cd246

1. The problem statement, all variables and given/known data
/int (2t+4)^-1/2 dt. the answer is 2(sqrt3)-2

2. Relevant equations

3. The attempt at a solution
u^-1/2*(1/2)du
(-1/2)(u^1/2)
(-1/4)(2t+4)^1/2
(-1/4)(sqrt 12)= (-1/4)(4(sqrt 3)/1)=sqrt 3
(sqrt 3)-1/2
2(sqrt 3)-1

2. Aug 10, 2007

rootX

-You don't need substitution for this.
simply use this exp:

$$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}$$

3. Aug 10, 2007

cd246

I did this: u^(1/2)/(1/2).
2u^(1/2)
2(2t+4)^(1/2)
(4t+8)^1/2
(12t)^(1/2)
2(3)^(1/2)
I believe I am further off than the first time.

4. Aug 11, 2007

CompuChip

If you don't see how to do the integral with rootX's suggestion, you can turn it around.
What happens if you differentiate $$x^{n + 1}/(n + 1)$$ with $x = (2t + 4)$ and $n = -1/2$?
Can you find a primitive now? (Watch the chain rule, you might need to add some constants to get the integrand back correctly)

5. Aug 11, 2007

malawi_glenn

What are the integration limits? From what to what are the integration performed over?

6. Aug 11, 2007

HallsofIvy

Well, no, that is not the answer! The answer is (2t+4)1/2+ C where C can be any constant. Is it possible that your integral had limits that you haven't told us?

Please, please, pleas, tell us what you are doing! This makes no sense until you tell us that you are using the substitution u= 2x- 4 so that (2x-4)-1/2 becomes u-1/2 (not 1/2) and 2dx= du so dx= (1/2)du
?? The anti-derivative of un is 1/(n+1) un+1 (as long as n is not -1). For n= -1/2, n+1= 1/2 and that becomes 2 u1/2, Multiplying by 1/2 from the "dx= (1/2) du" gives u1/2+ C= (2t+4)1/2+ C.

7. Aug 11, 2007

cd246

I cannot believe this, same problem but it is /int 4_0. I'm sorry i missed that.

To answer the question of what i am doing, first I have to figure out if this is something i can use the fund. rules of integrals or something i have to (or is preferred) substitute the function. For this one, i believe that i have to use sub.

Last edited: Aug 11, 2007
8. Aug 12, 2007

malawi_glenn

And yes people here have showing you that make the subsitute:

u = 2t + 4

Makes the problem slightly easier, have you tried this? And You do not seem to understand the basic algebraic rules here:

2(2t+4)^(1/2)
(4t+8)^1/2

Now this is true:
$$a^{1/2} * b^{1/2} = (ab)^{1/2}$$
So:
$$2(2t+4)^{1/2} = (4^{1/2})(2t+4)^{1/2} =(8t+16)^{1/2}$$

Anyway, if you make that substitution u, how does you new integral looks like?

9. Aug 12, 2007

HallsofIvy

I think rootX's point:
is that for simple linear substitutions, ax+ b, you should be able to just think: do the integral ignoring the linear part inside the power, and the divide by a: The anti-derivative of (ax+ b)n is (n/a)(ax+ b)n+1[/sup]. You don't really need to write out the substitution. That comes with experience.

10. Aug 12, 2007

cd246

i finally found my error. I understand the basics(for the most part) of integration.
It is the substitution and the more sophisticated equations that get me.