Integration Substitution

[TheFallen018]

Hi,

I seem to be having a bit of trouble with this question I came up against. I seem to be lacking in my intuition when it comes to u substitution in integrals. That's probably the root of the problem. However, I'm not quite sure what this question is asking anyways. If you guys could steer me in the right direction, I would be very grateful.
View attachment 9269

I've tried doing multiple things, but I think I just got to the point where I was confusing myself, and not getting anywhere.

 

[TheFallen018]

Hi, I've got this problem I've been trying to do. My understanding of u substation in integrals is still somewhat lacking, and I think that's where most of my troubles are coming in. I'm not really even sure what this question is asking, so any help would be greatly appreciated.



I've already tried a couple of ways to do it, but they seem to be wrong. I've gotten to the point where everything I'm doing is just confusing me more, so if you can help clear some things up, that would be amazing. Thanks.

 

[Deleted member 4993]

Hi, I've got this problem I've been trying to do. My understanding of u substation in integrals is still somewhat lacking, and I think that's where most of my troubles are coming in. I'm not really even sure what this question is asking, so any help would be greatly appreciated.

View attachment 9270

I've already tried a couple of ways to do it, but they seem to be wrong. I've gotten to the point where everything I'm doing is just confusing me more, so if you can help clear some things up, that would be amazing. Thanks.

View attachment 9270
\(\displaystyle \displaystyle{\sqrt{25 - e^{10x}}}\)

=\(\displaystyle \displaystyle{\sqrt{5^2 - \left({e^{5x}}\right )^2}}\)

=\(\displaystyle \displaystyle{5\sqrt{1 - \left(\dfrac{{e^{5x}}}{5}\right )^2}}\)

continue....

 

[Steven G]

Hi, I've got this problem I've been trying to do. My understanding of u substation in integrals is still somewhat lacking, and I think that's where most of my troubles are coming in. I'm not really even sure what this question is asking, so any help would be greatly appreciated.

View attachment 9270

I've already tried a couple of ways to do it, but they seem to be wrong. I've gotten to the point where everything I'm doing is just confusing me more, so if you can help clear some things up, that would be amazing. Thanks.

Personally I would use u=e^5x, but I understand you were asked to use another substitution

 

[sinx]

Hi, I've got this problem I've been trying to do. My understanding of u substation in integrals is still somewhat lacking, and I think that's where most of my troubles are coming in. I'm not really even sure what this question is asking, so any help would be greatly appreciated.

View attachment 9270

I've already tried a couple of ways to do it, but they seem to be wrong. I've gotten to the point where everything I'm doing is just confusing me more, so if you can help clear some things up, that would be amazing. Thanks.

what the expression says is y=summa (dy/dx)dx; in other words, y=[the integral of its derivative]. [it always is.]
Since the expression is too complicated to integrate as is, hopefully we can simplify it and do the integration. We can simplify through a substitution.
they are suggesting that a substitution of u=e^5x/5 will work, and are asking to
write the integrand as a function of u.
what this means is; between the summa sign and the du, this will be expressed in u, not x, [i.e. we will change the integrand from f(x) to f(u)].

so, you substitute u=e^5x/5
i.e. the numerator = e^5x, so e^5x=5u,
in the denominator you need to find what e^10x is as f(u),
e^10x=25[e^10x/25], =25u²
then simplify, and you got it.