Question

I have, yes... Another question regarding the evaluation of the integral of the following:

(sin(x^(1/2))/(x^(1/2)))dx, from x=(pi^2)/4 to pi^2

I am setting out to evaluate this integral as I have the rest of those I've done today- through u-substitution. So...
1. Can/should I set u = x^(1/2) ?
2. If not, then what is the appropriate/best thing to use as "u"?

Any and all help is greatly appreciated! :)

Answer 1

Yes, that u will work. Its best to use u-sub if you see a function and a form of its derivative. For yours
\[ u = \sqrt{x} \\\frac{du}{dx} = \frac{1}{2\sqrt{x}}\] which we see in the integral, except for the constant

Answer 2

Alright, that makes sense... I'm going to work through the problem:) will you be online/ monitoring this question for a while??? :D

Answer 3

I will for another half hour or so

Answer 4

Okay, so I have this so far, but am kind of confused as to the next step:

u = x^(1/2)
du = 1/2(x^(-1/2))dx

And...
integ.(sin(u))(u^-1)

Also... dx = du/(1/2(x^-1/2))
= (du/1)((2(x^(-1/2)))/1)
= (2x^(-1/2))du
So is the following also correct, considering "dx"?
2x^(-1/2)du = (2u)du (?)

Answer 5

Making the integral: (sin(u))(u^-1)(2u)du ... Does that "break rules"?

Answer 6

Hmm, here's what I would get step by step
\[u = \sqrt{x} \\
\frac{du}{dx} = \frac{1}{2\sqrt{x}} \to du=\frac{1}{2\sqrt{x}} dx \to 2du=\frac{1}{\sqrt{x}} dx \] Now we can replace our \[ \frac{1}{\sqrt{x}} dx\] to obtain
\[\int \frac{\sin{\sqrt{x}}}{sqrt{x}} dx \to \int \sin{u}\: 2du \to2\int \sin{u}\: du \]

Answer 7

Ahhh... Okay, I follow. :) why didn't I see that?! Lol...

Answer 8

Thank you! I see how that works out now :)

Answer 9

Not a problem, glad I could help

Answer 10

Oh, sorry, I know I closed the question, but after the step you posted above, when I remove the integral sign, and am taking the anti derivative, do I use the chain rule on theta (u), making it: 2(-cosu)u^2 ?

Answer 11

@kmeis002

Answer 12

:)

Answer 13

No need, now we are integrating with respect to u (du). The u-substitution is reversing the chain rule, so it takes care of it. The anti-differentive is just -cos(u) + C, then just plug in the sqrt(x)

Answer 14

Thank you!! :)

Answer 15

not a problem.