How do I Integrate this u substitution with limits.

In summary, the integral of e^x/ (1+e^2x) with limits ln 2 and 0 can be simplified by substituting u=e^x. The resulting integral is 1/e^2x which can be evaluated using the limits 2 and 1. However, the attempted solution using the derivative of arctan(u) is incorrect as the substitution cannot be made when there is a function of u in the integral. The correct substitution is u=tanθ.
  • #1
sg001
134
0

Homework Statement



Find ∫e^x/ (1+e^2x). dx , with limits ln 2 & 0
given u= e^x

Homework Equations





The Attempt at a Solution



u= e^x
du/dx = e^x
dx= du/e^x

sub limits of ln2 & 0 → u

Hence, limits 2 & 1

Therefore,

∫u* (1+e^2x)^-1* du/e^x

= ∫ u/ (u + e^3x)
= ∫ u/ e^3x
= ∫ 1/e^2x

= -e^-x
= -1/u
plugging in limits of 2 &1

Therefore, 0.2325...

Although i could not find this on the answer sheet did i do something wrong?
Please help, Thankyou.
 
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  • #2
Should it become this?
[itex]\int^{2}_{1}(1+u^{2})^{-1}du[/itex]
= [itex]\left[\frac{ln(1+u^{2})}{2u}\right]^{2}_{1}[/itex]
 
  • #3
th4450 said:
Should it become this?
[itex]\int^{2}_{1}(1+u^{2})^{-1}du[/itex]
Yes, this is correct. Letting [itex]u= e^x[/itex], [itex]du= e^xdx[/itex] so the numerator is just du and [itex](1+ e^{2x}[/itex] becomes [itex]1+ u^2[/itex]

= [itex]\left[\frac{ln(1+u^{2})}{2u}\right]^{2}_{1}[/itex]
However, this is incorrect. Yes, [itex]\int 1/u du= ln|u|+ C[/itex] but if you have a f(u) rather than u, you cannot just divide by f'(u)- that has to be already in the integral in order to make that substitution.

Instead, look up the derivative of arctan(u).
 
  • #4
Oops haha
should let u = tanθ
 

1. How do I choose the right substitution for my integration problem?

Choosing the right substitution often requires some trial and error. Generally, you want to choose a substitution that will eliminate the most complicated term in the integrand and leave you with a simpler expression to integrate. Common substitutions include u = x^2, u = sin(x), and u = e^x, but the best substitution will depend on the specific problem at hand.

2. Can I use a u substitution for definite integrals?

Yes, u substitutions can be used for both indefinite and definite integrals. When using a u substitution for a definite integral, be sure to also change the limits of integration to match the new variable.

3. What do I do with the constant when using a u substitution?

When using a u substitution, the constant is often ignored and carried along with the rest of the expression. However, if the constant is part of a term being substituted, it should be factored out and included in the final answer.

4. How do I know if I need to use a u substitution for integration?

There is no set rule for when a u substitution should be used, but it is often helpful when the integrand involves complicated functions or expressions. If you are having trouble integrating the original expression, a u substitution may be worth trying.

5. Can I use multiple u substitutions for one integration problem?

Yes, it is possible to use multiple u substitutions for one integration problem. In some cases, this may make the integration easier or more straightforward. However, be careful to keep track of all the substitutions and how they affect the limits of integration.

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