Integratre X(2X - 1)^1/2 using U = 2x - 1 as the substitute

[jak7774]

I have managed to work out that I need to sub in the U and replace dx with du, here is my workings:

Let U = 2X - 1
du/dx = 2
dx = du/2

Rearrange to find X
X = 1/2(U + 1)

Therefore

∫x(2x-1)^1/2 dx = ∫1/2(U+1)(U)^1/2 du/2 = 1/2 * 1/2 ∫(U + 1)(U)^1/2 du = 1/4∫U^3/2 + U^1/2 du

therefore 1/4[2U^5/2/5 + 2U^3/2/3] which simplifies to U^5/2/10 + U^3/2/6

but I the answer is this:
1/15(3x+1)(2x-1)^3/2 + c

How do I get to this result!?

 

[stapel]

I have managed to work out that I need to sub in the U and replace dx with du, here is my workings:

Let U = 2X - 1
du/dx = 2
dx = du/2

What are the specific relationships between X, U, x, and u?

What was the original exercise, complete with its instructions?

Thank you!

 

[Prove It]

I have managed to work out that I need to sub in the U and replace dx with du, here is my workings:

Let U = 2X - 1
du/dx = 2
dx = du/2

Rearrange to find X
X = 1/2(U + 1)

Therefore

∫x(2x-1)^1/2 dx = ∫1/2(U+1)(U)^1/2 du/2 = 1/2 * 1/2 ∫(U + 1)(U)^1/2 du = 1/4∫U^3/2 + U^1/2 du

therefore 1/4[2U^5/2/5 + 2U^3/2/3] which simplifies to U^5/2/10 + U^3/2/6

but I the answer is this:
1/15(3x+1)(2x-1)^3/2 + c

How do I get to this result!?

Now put it back as a function of x, and don't forget your +C. Then pull out any common factors.

 

[Steven G]

I have managed to work out that I need to sub in the U and replace dx with du, here is my workings:

Let U = 2X - 1
du/dx = 2
dx = du/2

Rearrange to find X
X = 1/2(U + 1)

Therefore

∫x(2x-1)^1/2 dx = ∫1/2(U+1)(U)^1/2 du/2 = 1/2 * 1/2 ∫(U + 1)(U)^1/2 du = 1/4∫U^3/2 + U^1/2 du

therefore 1/4[2U^5/2/5 + 2U^3/2/3] which simplifies to U^5/2/10 + U^3/2/6

but I the answer is this:
1/15(3x+1)(2x-1)^3/2 + c

How do I get to this result!?

You need a better understanding of what = means. One meaning is that whenever you see what is on the lhs of the = sign you can replace it with what is on the rhs of the = sign. You want x's and not u's? Then simply replace u with 2x - 1 since u= 2x - 1