Integrating: \frac{1}{(x^2+1)^2}

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In summary, the conversation discusses how to integrate the expression 1/(x^2+1)^2 and provides different methods for solving it, including partial fractions and trigonometric substitutions. The final answer is given as 1/2[x/(x^2+1)+arctan(x)]. The conversation also explains the process of choosing a trigonometric substitution and provides an example with a different expression in the denominator.
  • #1
sara_87
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Homework Statement



How do i integrate:

[tex]\int\frac{1}{(x^2+1)^2}[/tex]

?


Homework Equations



The answer is:

[tex]\frac{1}{4}(\frac{-2x}{x^2-1}-log(x-1)+log(x+1))[/tex]

The Attempt at a Solution



I don't know which method to use.
 
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  • #2


The answer you posted is correct if the integrand is 1/(1-x^2)^2. Is that what you meant? If so, try partial fractions.
 
  • #3


Im sorry, the question is correct but the answer is:

1/2[x/(x^2+1)+arctan(x)]

here partial fractions won't work because i will still have a term with (x^2+1)^2 in the denominator
 
  • #4


Have you tried a trig substitution? tan (theta) = x would be a good one to start with.
 
  • #5


sara_87 said:
Im sorry, the question is correct but the answer is:

1/2[x/(x^2+1)+arctan(x)]

here partial fractions won't work because i will still have a term with (x^2+1)^2 in the denominator

Ok, then it's a trig substitution. Like x=tan(t).
 
  • #6


how did you know what substitution to make??
 
  • #7


Draw a right triangle and label the sides and hypotenuse according to the expression in your integral. Since you have x^2 + 1, that corresponds to the hypotenuse. Label the side opposite your acute angle as x, label the base as 1, and label the hypotenuse as sqrt(x^2 + 1). So tan(theta) = x/1, and sec^2(theta)d(theta) = dx. Use these two equations to completely convert you integral from expressions in x and dx to ones in theta and d(theta).
 
  • #8


If you're asking how did we know to make a trig substitution, they are good bets when you have factors involving the sum or difference of squares, such as sqrt(x^2 + a^2), sqrt(a^2 - x^2), or sqrt(x^2 - a^2).
 
  • #9


what if we have (5+x^2)^2 in the denominator?
 
  • #10


Label the opposite side as x and the base as sqrt(x), so tan(theta) = x/sqrt(5)
 
  • #11


thank you
 

1. What is the function of "Integrating: \frac{1}{(x^2+1)^2}"?

The function of this expression is to find the integral of the given function, which is represented by the symbol "dx" in calculus. In other words, it helps determine the area under the curve of the given function.

2. How do you solve for the integral of \frac{1}{(x^2+1)^2}?

To solve for the integral of this expression, you can use various techniques such as substitution, integration by parts, or partial fractions. It is important to remember the basic rules of integration and choose the most suitable method for the given function.

3. What is the importance of integrating \frac{1}{(x^2+1)^2}?

The process of integration is important in mathematics and science as it allows us to calculate the area under a curve and solve complex problems involving rates of change. In the case of this specific expression, it is commonly used in physics, engineering, and economics to solve various problems.

4. Are there any specific conditions for solving the integral of \frac{1}{(x^2+1)^2}?

Yes, there are certain conditions that need to be met in order to solve the integral of this expression. For example, the function must be continuous on the given interval and the limits of integration must be defined. It is also important to check for any discontinuities or singularities that may affect the solution.

5. Can you provide a step-by-step guide for solving the integral of \frac{1}{(x^2+1)^2}?

Yes, here is a basic step-by-step guide for solving the integral of this expression using the substitution method:

  1. Let u = x^2 + 1 and du = 2x dx
  2. Rewrite the expression as: \frac{1}{(x^2+1)^2} = \frac{1}{u^2} \cdot \frac{1}{2x}
  3. Substitute u and du into the expression, and solve for the integral: \int \frac{1}{(x^2+1)^2} dx = \frac{1}{2} \int \frac{1}{u^2} du
  4. Using the power rule of integration, the solution becomes: \frac{1}{2} \cdot \frac{-1}{u} + C = -\frac{1}{2u} + C
  5. Substitute back u = x^2 + 1 and simplify the solution: -\frac{1}{2(x^2+1)} + C

Thus, the final answer for the integral of \frac{1}{(x^2+1)^2} is -\frac{1}{2(x^2+1)} + C.

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