How to integrate int x/(x^2+1)dx

  • Thread starter naspek
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In summary, the general approach to integrating x/(x^2+1)dx is to use the substitution method. The most common substitution for this integral is u = x^2+1, which allows for the use of known integration techniques. Trigonometric substitutions, such as u = tan(x), can also be used. The integral cannot be solved without substitution, and another method that can be used is partial fractions. However, this method may be more complex and time-consuming.
  • #1
naspek
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[tex]\int{\left(\frac{x}{x^2 + 1}\right)\,dx}[/tex]

how to integrate?
i know [tex]\int{\left(\frac{1}{x^2 + 1}\right)\,dx}[/tex]
is tan^-1 x + C

how am i going to start answering this question
 
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  • #2


try substituting u=x2+1, du=?
 
  • #3


rock.freak667 said:
try substituting u=x2+1, du=?

meaning.. i should u integration by substitution?
 
  • #4


Yes, try the u substitution.
 
  • #5


got it..
1/2 ln |x^2 + 1| + C

correct?
 
  • #6


That's right. One thing though: since x2 + 1 is always greater than 0, no absolute values are needed.
 

Related to How to integrate int x/(x^2+1)dx

1. What is the general approach to integrating x/(x^2+1)dx?

The general approach to integrating x/(x^2+1)dx is to use the substitution method. This involves substituting the expression inside the parentheses with a new variable, u, and rewriting the integral in terms of u. This allows for the use of known integration techniques, such as the power rule or integration by parts.

2. What substitution should I use for x/(x^2+1)dx?

The most common substitution for x/(x^2+1)dx is u = x^2+1. This allows for the integral to be rewritten as ∫(1/u)du, which can be easily integrated using the natural logarithm function.

3. Can I use trigonometric substitutions to integrate x/(x^2+1)dx?

Yes, x/(x^2+1)dx can also be integrated using trigonometric substitutions. One possible substitution is u = tan(x), which leads to the integral becoming ∫(1/u^2+1)du. This can be integrated using the inverse tangent function.

4. Can the integral of x/(x^2+1)dx be solved without substitution?

No, the integral x/(x^2+1)dx cannot be solved without substitution. This is because the expression x^2+1 does not have a known antiderivative, making it impossible to use basic integration techniques.

5. Are there any other methods for integrating x/(x^2+1)dx?

Yes, in addition to substitution and trigonometric substitutions, x/(x^2+1)dx can also be integrated using partial fractions. This involves breaking the fraction into smaller, simpler fractions and then integrating each term separately. However, this method can be more complex and time-consuming compared to the substitution method.

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