Integration by trig. substitution

In summary, the individual is seeking help with an exercise for an upcoming exam. They have provided a link to a scan of their work and are requesting feedback on what they did wrong. It is suggested that there may be a mistake with "u+1" and it should be "u+2" in the numerator. The individual also mentions a mistranslation of sec(t) to u when converting from theta back to x. They thank the person for their help.
  • #1
mad
65
0
Hello everyone
I have an exam tomorrow and I would really appreciate if someone could tell me what I did wrong with this exercice. I did it on paper and I scanned it. Here is the link to the scan:

The answer in the book is sqrt(x^2 + x +5/4) + 2ln(sqrt(x^2 + 2x + 2) + x +1) + C

http://img223.echo.cx/img223/8425/problem490012al.jpg

It is hosted on imageshack.us
download it and open it in windows so you can change the zoom size for better viewing

Thanks a lot in advance !
 
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  • #2
If u = x + 1, where does u + 1 come from? That is just the first thing I saw, it may lead to your other problems.
 
  • #3
It should be "u+2" in the numerator after doing the first sub.

Daniel.
 
  • #4
dextercioby said:
It should be "u+2" in the numerator after doing the first sub.

Daniel.

Woops. Was this the only problem? I think this only changes the arctg from the final answer and not the ln(..) of the answer. I hate when this type of error happens..
 
Last edited:
  • #5
When translating from theta back to x, you mistranslated sec(t).

Next to the triangle you have

[tex] sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{u} [/tex]

cos(theta) is not u, try looking at it again.
 
  • #6
whozum said:
When translating from theta back to x, you mistranslated sec(t).

Next to the triangle you have

[tex] sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{u} [/tex]

cos(theta) is not u, try looking at it again.

Thanks a lot :)
 

Related to Integration by trig. substitution

1. What is the purpose of using trigonometric substitution in integration?

Trigonometric substitution is used to simplify integrals that involve expressions with radicals or powers of trigonometric functions. By substituting a trigonometric expression for a variable, we can transform the integral into a more manageable form that can be easily solved.

2. How do you know when to use trigonometric substitution in an integral?

Trigonometric substitution is usually used when the integral involves expressions with radicals, especially if the power of the radical is odd. It can also be used when the integral involves expressions with squares of trigonometric functions.

3. What are the common trigonometric substitutions used in integration?

The most commonly used trigonometric substitutions are:

  • For expressions involving √(a^2 - x^2), we use x = a sinθ.
  • For expressions involving √(a^2 + x^2), we use x = a tanθ.
  • For expressions involving √(x^2 - a^2), we use x = a secθ.

4. Can any integral be solved using trigonometric substitution?

No, not every integral can be solved using trigonometric substitution. It is only useful when the integral involves expressions with radicals or powers of trigonometric functions. Other techniques, such as u-substitution or integration by parts, may be needed for other types of integrals.

5. Are there any special cases to consider when using trigonometric substitution?

Yes, there are a few special cases to consider when using trigonometric substitution. These include:

  • The substitution may only work for a specific range of values for the variable.
  • The integral may need to be split into multiple parts to be solved.
  • Care must be taken when dealing with trigonometric identities and simplifying the substituted expression.

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