Trigonometric Substitution

In summary, the student is trying to solve a problem involving the "e" symbol and the "x" in the exponent and is having difficulty. He starts by setting sin(theta) equal to e^x and cos(theta) equal to sqrt(1-e^2x), and then uses those values to solve for sin^3(theta)/cos((theta). He ends up with sin^2/2 + c, which looks correct to him. However, he is missing a bracket in tan, and he needs to substitute for dx in the integral.
  • #1
protivakid
17
0

Homework Statement



[tex]\int[/tex] [tex]\frac{e^{3x}dx}{\sqrt{1-e^{2x}}}[/tex]


Homework Equations





The Attempt at a Solution



Alright so I am able to do other similar problems fine, I think it is the "e" that is throwing me off as well as the fact that the "x" is in the exponent. I started the problem as so...

sin[tex]\vartheta[/tex]=[tex]\frac{e^{\sqrt{x}}}{\sqrt{1}}[/tex]

cos[tex]\vartheta[/tex]d[tex]\vartheta[/tex]=[tex]\frac{e^{\sqrt{x}}dx}{\sqrt{1}}[/tex]

Am I off to the right start and if so can I have some helpful hints as what to do next? Thanks guys.

By the way, those symbols next to Sin & Cos are Theta, I am still learning how to use this forum, sorry.
 
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  • #2
I think you should have sin(theta) = e^x, cos(theta) =sqrt(1-e^2x). Remember that for any n, (e^nx) =(e^x)^n
 
  • #3
Re-write it.

[tex]\int\frac{(e^x)^3dx}{\sqrt{1-(e^x)^2}}[/tex]

[tex]e^x=\sin xdx[/tex]

Take the natural log, then it's derivative.
 
  • #4
Alright so taking your advice I set sin(theta) to e^x dx, and cos(theta) to sqrt(1-e^2x). That then gave me sin^3(theta)/cos((theta). I set u=sin(theta)d(theta) and du=cos(theta) which gave me u^3du^-1. That then became u^2/2 which is sin^2/2. My final answer U then got from that was e^2x/2 + c. Does that sound correct, if not please advise me and I am sorry for misunderstanding your help.
 
  • #5
Looks wrong.
 
  • #6
You cannot have a du in the denominator. Think about what this would means in terms of the Riemann sum and you'll see that it doesn't make any sense. Instead, consider an expansion of sin(x)³. That is

[tex] \frac{sin^3(x)}{cos(x)} = sin(x) \frac{1-cos^2(x)}{cos(x)} = tan(x) - sin(x)cos(x) [/tex] You can integrate this.
 
  • #7
I am trying to set up a triangle for visual aid, is the following correct...

sin=e^x
cos=sqrt(1-e^2x)
tan= (e^x)/(sqrt(1-e^2x)

Thanks guys, greatly appreciated.
 
  • #8
Yes, those are correct, though you're missing a bracket in tan
 
  • #9
Thanks, i'll try to take it from here but I don't think there is too much left to do.
 
  • #10
I didn't read all the way back,and that was my mistake, but did you remember to substitute for dx in the integral as well?

If [itex] e^x = \sin\theta [/itex] then [itex] e^x dx = \cos\theta d\theta [/itex] which implies that [itex] dx = \mathrm{cot}\theta d\theta [/itex]. That should actually make the integral quite easy
 

1. What is Trigonometric Substitution?

Trigonometric Substitution is a technique used in calculus to simplify the evaluation of integrals involving algebraic expressions with square roots or quadratic equations.

2. When is Trigonometric Substitution used?

Trigonometric Substitution is typically used when the integrand (function being integrated) contains an expression in the form of √(a^2 - x^2) or √(x^2 + a^2), where a is a constant.

3. How does Trigonometric Substitution work?

Trigonometric Substitution involves substituting a trigonometric function (such as sine, cosine, or tangent) for the variable in the integral. This allows us to use trigonometric identities and properties to simplify the integral and solve it.

4. What are the three main types of Trigonometric Substitutions?

The three main types of Trigonometric Substitutions are:
1. √(a^2 - x^2) - this involves using the substitution x = a sinθ
2. √(x^2 + a^2) - this involves using the substitution x = a tanθ
3. √(x^2 - a^2) - this involves using the substitution x = a secθ

5. What are some common mistakes to avoid when using Trigonometric Substitution?

Some common mistakes to avoid when using Trigonometric Substitution are:
- Forgetting to substitute the limits of integration
- Forgetting to convert the trigonometric function back to the original variable
- Using the wrong trigonometric substitution for the given integral
- Making mistakes while simplifying the integral using trigonometric identities

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