what is the Integration of x.y.[(x^Integration of x.y.[(x^2 +y^2 +25)^3/2 ] dx.dy from (0<x<1) &(0<y<1) ?
To find the double integral of the given expression, we will break it down into two steps: first, we will integrate with respect to x, and then integrate the result with respect to y.
Step 1: Integration with respect to x
To integrate the expression with respect to x, we treat y as a constant and integrate the expression (x^2 + y^2 + 25)^(3/2) with respect to x. Let's denote this integral as I1.
To simplify the integration, let's make a substitution. Let u = x^2 + y^2 + 25.
Now, differentiate both sides of the equation with respect to x:
du/dx = 2x.
Rearranging the equation, we have dx = du / (2x).
Making this substitution, the integral I1 becomes:
∫ (x^2 + y^2 + 25)^(3/2) dx
= ∫ (u)^(3/2) (du / 2x)
= (1/2) ∫ u^(3/2) / x du.
At this point, we can treat x as a constant, and integrate u^(3/2) with respect to u:
(1/2) ∫ u^(3/2) du
= (1/2) * (2/5) * (u)^(5/2) + C
= (1/5) u^(5/2) + C.
So, the result of the integration with respect to x is (1/5) (x^2 + y^2 + 25)^(5/2) + C. Let's denote this as I2.
Step 2: Integration with respect to y
Now, we need to integrate I2 with respect to y. Treating x as a constant, we integrate (1/5) (x^2 + y^2 + 25)^(5/2) with respect to y. Let's denote this integral as I3.
Since we are integrating with respect to y, the x^2 and 25 terms act as constants. Therefore, the integral reduces to:
(1/5) (x^2 + 25)^(5/2) ∫ (y^2 + 1)^(5/2) dy.
We can solve this integral using a basic trigonometric substitution. Let z = y^2 + 1. Differentiating both sides gives dz = 2y dy. Rearranging the equation and substituting, we have dy = dz / (2y).
Making this substitution, the integral becomes:
(1/5) (x^2 + 25)^(5/2) ∫ (z)^(5/2) (dz / (2y))
= (1/10y) (x^2 + 25)^(5/2) ∫ (z)^(5/2) dz.
Integrating z^(5/2) with respect to z gives:
(1/10y) (x^2 + 25)^(5/2) * (2/7) * (z)^(7/2) + C
= (1/35y) (x^2 + 25)^(5/2) * (z)^(7/2) + C.
Substituting back y^2 + 1 for z, we have:
(1/35y) (x^2 + 25)^(5/2) * (y^2 + 1)^(7/2) + C.
So, the final result of the double integral is:
(1/35) * (x^2 + 25)^(5/2) * (y^2 + 1)^(7/2) + C.
Please note that C is the constant of integration and should be determined based on the given limits of integration or any additional information provided in the question.