Solve Integration Problem: \int_{1}^{2}(\frac{2}{2x+1})^3 dx

[disregardthat]

Homework Statement

$$\int_{1}^{2}(\frac{2}{2x+1})^3 dx$$

Homework Equations

Normal integral equations

The Attempt at a Solution

$$\int_{1}^{2}(\frac{2}{2x+1})^3 dx = 2^3\int_{1}^{2}\frac{1}{(2x+1)^3} dx$$

u=2x+1

$$2^3\int_{1}^{2}\frac{1}{u^3} dx = 2^3\int_{1}^{2}u^{-3} dx$$

Antiderivate of $$u^{-3} = \frac{u^{-3+1}}{-3+1} = \frac{u^{-2}}{-2}= \frac{1}{-2u^2}$$

Plotting the real u in: $$\frac{1}{-2(2x+1)^2}$$

$$2^3\int_{1}^{2}\frac{1}{-2(2x+1)^2} = 2^3\left(\frac{1}{-2(2x+1)^2}\right)_1^2 =2^3\left(\frac{1}{-2(2 \cdot 2+1)^2}\right) - 2^3\left(\frac{1}{-2(2 \cdot 1+1)^2}\right) = \left(\frac{2^3}{-50}\right) - \left(\frac{2^3}{-18}\right)$$

$$= \left(\frac{72}{-450}\right) - \left(\frac{200}{-450}\right) = \left(\frac{72-200}{-450}\right) = \left(\frac{-128}{-450}\right) = \frac{128}{450} = \frac{64}{225}$$

This gives about: 0.284444...

When I type this function on the calculator, the area between x=1 and x=2 is 0.1422222...

That is half of my answer, what have I done wrong?

 

[happyg1]

u=2x+1
du=2dx
you need u du not u dx up there in your substitution.
So you have to divide by 2 outside the integral.

 

[disregardthat]

I see, I have to divide by two.

But i did not understand why.

EDIT: I mean, how do you find du?

 

[happyg1]

Hey,
It's just the derivative of 2x+1 with respect to x, which is 2 dx. You always need to find du when you use the u substitution.

 

[disregardthat]

All right.
So how am I to set it up then?

$$\int_{1}^{2}(\frac{2}{2x+1})^3 dx =$$

Any of these?

$$\frac{d}{dx}2^3\int_{1}^{2}\frac{1}{(u)^3} dx$$

$$\frac{2^3}{du}\int_{1}^{2}\frac{1}{(u)^3} dx$$

Do you have an example of an solution to a task like this? I want to know how you should set up the solution so I see what is really going on.

 

[happyg1]

Hi,
$$\int_{1}^{2}(\frac{2}{2x+1})^3 dx = 2^3\int_{1}^{2}\frac{1}{(2x+1)^3} dx$$

u=2x+1
du=2dx

$$\frac{2^3}{2}\int_{1}^{2}\frac{1}{u^3} du = \frac{2^3}{2}\int_{1}^{2}u^{-3} du$$

You just have to make sure that you've got $$u du$$ not $$u dx$$. The derivative of the u=2x+1 has to be included. You don't want an extra 2 in there, so you just divide it out. That's all. You integrated this perfectly, just missed the du substitution.
Hope this helps.
CC

 

[happyg1]

or maybe more clear
u=2x+1
du=2dx
$$\frac{1}{2}du=dx$$
then factor out the 1/2.

 

[disregardthat]

Oh, right. So when I now have the du after the u^-3 then I just antiderivate just like before right?
the du in the end doesn't have any meaning to how I am supposed to antiderivate the function, right? Without of course the division of 2^3.

Uhm, I mean:

When you have a function, and use substition, you have to find the derivative of the substibtution and then divide the function with it, (it is enough to divide the 2^3) and then just antiderivate as before, only with a du instead of dx. Is that correct?

And should the du at the end stand there until i have antiderivated u or when i put 2x+1 back again?

 

[happyg1]

When you have a function, and use substition, you have to find the derivative of the substibtution and then divide the function with it, (it is enough to divide the 2^3) and then just antiderivate as before, only with a du instead of dx. Is that correct?

Correct. When you integrate, the du goes away. Then you put back in your u, just as you did.

 

[disregardthat]

All right, thanks for your help