Integrating -2xy/(1+x^2): Solution

In summary, the conversation is about finding the equation of a curve in the xy plane that passes through a given point and has a specific slope at any given point. The question is incomplete and the person answering requests for more information. The equation given is a first order ODE and the person suggests using variable separable method to solve it, but also mentions that the given information about the curve passing through a specific point has not been used yet.
  • #1
lost_2
5
0
HI there! i was wondering if you could help me with this..

how do you integrate -2xy/(1+x^2)?

Thank you.
 
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  • #2
lost_2 said:
HI there! i was wondering if you could help me with this..

how do you integrate -2xy/(1+x^2)?

Thank you.
Welcome to the forums,

Your question is incomplete. What are you integrating with respect to? Is y a function of x or vice versa?

As an aside, feel free to post a new topic for each new question, you don't need to tag your question on the end of others/
 
  • #3
Hey!

Thanks for the quick reply.. Coz it a similar qns so i posted together with this. That equation is the gradient of a curve. So it's just to integrate with respect to x? How do you go about doing it?

Thank you.
 
  • #4
Anyways this is the whole qns.

find the equation of the curve in the xy plane that passes through (1,2) and whose tangent line at a point (x,y) has a slope of -2xy/(1+x^2).
 
  • #5
lost_2 said:
Anyways this is the whole qns.

find the equation of the curve in the xy plane that passes through (1,2) and whose tangent line at a point (x,y) has a slope of -2xy/(1+x^2).
Thanks, it's always a good idea to post the entire question. So you know that,

[tex]\frac{dy}{dx} = -\frac{2xy}{1+x^2}[/tex]

Which is a first order ODE. What methods do you know for solving first order ODE's?
 
  • #6
by using variable separable method?
 
  • #7
is the answer y(1+x^2)= c?
 
  • #8
You have not yet used the fact that the curve passes through (1, 2).
 

1. What does "integrating" mean in this context?

Integrating refers to the process of finding the antiderivative of a given function. Essentially, it involves finding a function whose derivative is equal to the given function.

2. What does the notation "-2xy/(1+x^2)" represent?

The notation -2xy/(1+x^2) represents a function that includes variables x and y and is divided by the expression 1+x^2. This type of function is known as a rational function.

3. How do you solve for the antiderivative of -2xy/(1+x^2)?

To solve for the antiderivative, we use the rules of integration, which include power rule, product rule, and chain rule. We also use the techniques of substitution and integration by parts to simplify the function and find its antiderivative.

4. Can you explain the steps for integrating -2xy/(1+x^2)?

First, we split the rational function into two separate fractions: -2xy and 1/(1+x^2). Next, we use the substitution technique by assigning u = 1+x^2 to the second fraction. Then, we apply the power rule and product rule to find the antiderivative of each fraction. Finally, we substitute back the value of u and simplify the expression to get the final solution.

5. What is the final solution for integrating -2xy/(1+x^2)?

The final solution is given by: -xy - ln|1+x^2| + C, where C is the constant of integration. This solution can also be written as -xy - ln(1+x^2) + C, where ln represents the natural logarithm function.

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