How to evaluate this definite integral?

[nashsth]

Hello, I don't know how to approach this problem, provided in the image below:

I am assuming that in order to solve this problem, you have to transform the 2nd integral into the form of the first integral, but I am not sure if that's even the way to solve it, and even if it was, I don't know how to transform it. Any help would be appreciated.

Thanks :-)

 

[SteamKing]

Hello, I don't know how to approach this problem, provided in the image below:

View attachment 92273

I am assuming that in order to solve this problem, you have to transform the 2nd integral into the form of the first integral, but I am not sure if that's even the way to solve it, and even if it was, I don't know how to transform it. Any help would be appreciated.

Thanks :-)

Have you studied u-substitution for indefinite integration yet?

 

[nashsth]

Have you studied u-substitution for indefinite integration yet?

We learned about it a few days ago... I don't know what to substitute to solve the problem :-(

 

[nashsth]

We learned about it a few days ago... I don't know what to substitute to solve the problem :-(

Ohh ok I see I see

so you let u = sqrt(x), then x = u^2, so that du = 2u du

Then you plug them into the 2nd integral, and then you get integral of f(u) / u * 2u du, which then simplifies into 2 * integral of f(u) du, which is 3, as given by the first integral, so that the answer is 2*3 = 6.

Is this right? O_O

 

[Mark44]

Ohh ok I see I see

so you let u = sqrt(x), then x = u^2, so that du = 2u du

So dx = 2udu...

Then you plug them into the 2nd integral, and then you get integral of f(u) / u * 2u du, which then simplifies into 2 * integral of f(u) du, which is 3, as given by the first integral, so that the answer is 2*3 = 6.

Is this right? O_O

Yes

In the future, please post questions like this one in the Homework & Coursework section, not here in the math technical sections.

 

[nashsth]

So dx = 2udu...
Yes

In the future, please post questions like this one in the Homework & Coursework section, not here in the math technical sections.

Thanks for your help :-) And yes I will make sure to post similar questions to the appropriate sections. Apologies for the inconvenience