change of variables

[Domibellic]

hi guys, i solved this integral with two different ways, but got different answers(after replace the X with a number). Someone can explain why one of them isn't correct?

 

[HallsofIvy]

In the first one, if you are letting \(\displaystyle t= e^x\) then \(\displaystyle e^{3x}\) is \(\displaystyle t^3\), NOT \(\displaystyle t^2\)!

 

[Domibellic]

In the first one, if you are letting \(\displaystyle t= e^x\) then \(\displaystyle e^{3x}\) is \(\displaystyle t^3\), NOT \(\displaystyle t^2\)!

when replacing the dx, to dt/(e^x)-->t^3/t^1=t^2

 

[Deleted member 4993]

The work above is correct.



The work below is NOT correct.



Here the integration is NOT correct. In the denominator of the integrand, you have [t^1/3]².

For the integration to be "somewhat" correct - the numerator needs to be [t^-2/3]dt - instead of just "dt".

 

[MarkFL]

We are given:

[MATH]I=\int\frac{e^{3x}}{1+e^{2x}}\,dx[/MATH]
If I were going to use the second substitution you posted, I would let:

[MATH]t=e^{3x}\implies dt=3e^{3x}dx[/MATH]
And now we have:

[MATH]I=\frac{1}{3}\int\frac{1}{1+t^{\frac{2}{3}}}\,dt=\frac{1}{3}\int\frac{t^{-\frac{2}{3}}}{t^{-\frac{2}{3}}+1}\,dt[/MATH]
Next, let:

[MATH]u=t^{\frac{1}{3}}\implies du=\frac{1}{3}t^{-\frac{2}{3}}\,dt[/MATH]
And we have:

[MATH]I=\int\frac{1}{u^{-2}+1}\,du=\int\frac{u^2}{u^{2}+1}\,du=\int 1-\frac{1}{u^2+1}\,du[/MATH]
[MATH]I=u-\arctan(u)+C=e^x-\arctan(e^x)+C[/MATH]

 

[Steven G]

In the first one, if you are letting \(\displaystyle t= e^x\) then \(\displaystyle e^{3x}\) is \(\displaystyle t^3\), NOT \(\displaystyle t^2\)!

I do agree with everything you said but the OP broke off an e^x to use for e^xdx = dt. So in the end the OP just needed to rewrite e^(2x) in terms of t

 

[Steven G]

yes, you did make a mistake but i am concerned that you said that i solved this integral with two different ways, but got different answers(after replace the X with a number).

Just because you got different results plugging in an x-value does NOT mean anything. The two results you got could differ by a constant!

Integrate (x+2)^2 using the substitution u=(x+2) and integrate (x^2+2x+4) directly and see if you get the same answer.

 

[Domibellic]

Thank you all!