can you help me find
integration [2xy/(x^2+y^2)^2] do x
?
∫ 2 x y dx / ( x² + y² )² =
2 y ∙ ∫ x ∙ dx / ( x² + y² )² =
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Apply substitution:
u = x² + y²
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2 y ∙ ∫ x ∙ dx / u² =
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Derivative:
du / dx = d ( x² ) / dx + d ( y² ) / dx
du / dx = 2 x + 0
du / dx = 2 x
du = 2 x dx
2 x dx = du
x dx = du / 2
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= 2 y ∙ ∫ ( du / 2 ) / u² =
= ( 2 y / 2 ) ∙ ∫ du / u² =
= y ∙ ∫ du / u² =
y ∙ ∫ u⁻² ∙ du / =
y ∙ u ⁻ ² ⁺ ¹ / ( - 2 + 1 ) + C
y ∙ u ⁻ ¹ / ( - 1 ) + C =
y ∙ ( - 1 / u ) + C =
- y / u + C =
- y / ( x² + y² ) + C
Sure, I can help you with finding the integration! But before we get started, let me warn you: integrating can be a bit challenging, so it's important to keep a positive attitude. Now, let's dive in!
To integrate the expression [2xy/(x^2+y^2)^2] dx, we can make use of a technique called substitution. Let's assume u = x^2 + y^2. Therefore, du/dx = 2x.
Now, let's rearrange the given expression a bit:
[2xy/(x^2+y^2)^2] dx = [xy/(x^2+y^2)(x^2+y^2)] dx = (y/u^2) du
Now, we just have to integrate (y/u^2) du. Remember, we can treat y as a constant! So, we pull it out of the integral sign:
(integral of y/u^2) du = y integral of u^(-2) du = -y/u + C
Finally, replacing u with x^2 + y^2, we get the final answer as:
-integral of (1/(x^2+y^2)) dx + C
Don't worry if this solution seems a bit complicated – integration can be a bit of a circus act! But I hope this clownish explanation helped you find your way through it.
Sure! To find the integration of 2xy/(x^2+y^2)^2 with respect to x, we will use the technique of substitution.
Let's begin by performing the substitution:
Let u = x^2 + y^2.
Differentiating both sides of the equation with respect to x, we get:
du/dx = 2x.
Rearranging, we find:
x = (1/2)(du/dx)
Now, we can express the given integral in terms of u:
∫ [2xy/(x^2+y^2)^2] dx = ∫ [(2y/u^2)] (1/2)(du/dx) dx.
Rearranging, we have:
∫ [(2y/u^2)] (1/2)(du/dx) dx = (1/2) ∫ (2y/u^2) du.
Simplifying, we obtain:
(1/2) ∫ (2y/u^2) du = ∫ (y/u^2) du.
Now, integrating with respect to u, we have:
∫ (y/u^2) du = y ∫ (1/u^2) du.
Evaluating the integral, we get:
y ∫ (1/u^2) du = - (y/u) + C,
where C is the constant of integration.
Finally, substituting back x = (√u)/2 into the equation, we can express the final answer in terms of x:
- (y/√(x^2 + y^2)) + C.
Therefore, the indefinite integral of 2xy/(x^2+y^2)^2 with respect to x is:
- (y/√(x^2 + y^2)) + C.
Of course! I can help you find the integration of the given expression with respect to variable x. Let's break down the steps on how to solve this.
Step 1: We have the integral of 2xy / (x^2 + y^2)^2 with respect to x. Treat y as a constant for now.
Step 2: To solve this integral, we can use a substitution. Let's substitute u = x^2 + y^2. This implies du/dx = 2x, and solving for dx gives us dx = du / (2x).
Step 3: Substitute these values into the integral to obtain the new integral:
∫ (2xy / (x^2 + y^2)^2) dx = ∫ (2xy / u^2)(du / (2x)).
Step 4: Simplify the expression to get ∫ (y / u^2) du.
Step 5: Now we have a simple integral to solve. The integration of (y / u^2) du can be calculated by applying the power rule for integration, which states that the integral of x^n dx equals (x^(n+1))/(n+1).
Step 6: Applying the power rule, we get ∫ (y / u^2) du = y ∫ (1 / u^2) du = y * (-1 / u) + C, where C is the constant of integration.
Step 7: Substitute our original variable x back into the expression, which means replacing u with x^2 + y^2:
y * (-1 / (x^2 + y^2)) + C.
Therefore, the solution to the given integration is -y / (x^2 + y^2) + C, where C is a constant of integration.