Is ∫(1/(x²+25))dx² Correctly Evaluated as log(x²+25)?

[labinojha]

My friends were discussing about this problem (which they made up themselves).
∫\frac{1}{x^{2}+25}dx^{2}
They were substituting x^2 for y (x^{2}=y) and thus the answer would come to be log(y+25)
that is log(x^{2}+25)

I don't think this is the case , i guess that we would be differentiating wrt a 2nd degree curve like a parabola in case of this problem .

Would you people point out what's the real thing.

 

[Mute]

If the integral is really as you wrote it,

\int d(x^2) \frac{1}{x^2+25},

then you are free to set y = x^2, as in this case your differential is d(x^2), and it already contains the x^2 so you are free to make the substitution.

If the integral were

\int dx \frac{1}{x^2+25},

then you can of course still substitute y = x^2, but then the differential term is different: dx \rightarrow dy/(2x) = dy/(\pm 2\sqrt{y}). Note that if this were a definite integral were the limits went from some negative value to a positive one, you would have to split the integral into two pieces, one from the negative value to 0 and one from 0 to the positive value, and then make the substitution, as you need to choose a sign for the square root and it's different for x > 0 and x < 0.

 

[HallsofIvy]

In general, if g(x) is a differentiable function, then dg(x)= g'(x)dx.

d(x^2)= 2xdx so
\int\frac{1}{x^2+ 25}d(x^2)= \int\frac{2x}{x^2+ 25}dx

Now, if you let u= x^2+ 25, du= 2xdx and the integral becomes
\int\frac{du}{u}= ln(u)+ C= ln(x^2+ 25)+ C
just as before.

That really is the case.

 

[labinojha]

Thanks for the replies . My teacher also did the same substitution process when i asked him about this.

These integrations and differentiations are done in with respect to the x-coordinate which is a straight line .
I was wondering if we could carry out these operations with respect to curves as well like the x^{2}.

Sorry if this is absurd, but i would be glad if i was clarified .

 

[HallsofIvy]

Yes, you integrate a function f(x) with respect to another function g(x). You are, in effect, "changing the scale". For the Riemann integral, such a function, g, would have to be differentiable and, as I said before, \int f(x)dg(x)= \int f(x)g&#039;(x)dx[/tex]. For the Stieljes integral, g does not have to differentiable, only increasing.