what is the Integration of x.y.[(x^2 +y^2 +25)^3/2 ] dx.dy from (0<x<1) &(0<y<1) ?

?[0,1]?[0,1] (x^2+y^2+25)^(3/2) dx dy = 130.046

This is not a trivial task, since it involves trig substitutions. Anyway, look here to find the first indefinite integral. Plug in your two y-values, giving a similar integral involving only x.

http://www.wolframalpha.com/input/?i=%E2%88%AB(x%5E2%2By%5E2%2B25)%5E(3%2F2)+dy

To find the integration of the given expression, we will perform a double integration over the given limits.

Let's start by integrating with respect to x:

∫[0 to 1] ∫[0 to 1] x · y · [(x^2 + y^2 + 25)^(3/2)] dx dy

First, let's simplify the expression inside the integral by expanding the power:

∫[0 to 1] ∫[0 to 1] x · y · [(x^2 + y^2 + 25)^(3/2)] dx dy

Let's distribute the x and y terms:

∫[0 to 1] ∫[0 to 1] [x^3y + xy^3 + 25xy] dx dy

Next, let's integrate with respect to x:

∫[0 to 1] [x^4/4y + x^2y^3/2 + 25/2xy^2] evaluated from 0 to 1 dy

Simplifying the expression:

∫[0 to 1] [(1/4y) + (y^3/2)/2 + (25/2)y^2] dy

Integrating this expression with respect to y:

(1/4)ln|y| + (y^5/20)/2 + (25/6)y^3 evaluated from 0 to 1

Simplifying further:

(1/4)ln(1) + (1/40) + (25/6) - (1/4)ln(0) - (0/40) - (25/6) = (1/4)ln(1) + (1/40) + (25/6) - (1/4)ln(0) - (0/40) - (25/6)

Since ln(0) is undefined, the integral is not defined.

Therefore, the integration of the given expression is undefined over the given limits.

To find the integration of the given function, x.y.[(x^2 + y^2 + 25)^3/2], over the given limits of x and y, you will need to perform a double integration.

Let's start with the inner integration. We'll integrate the function with respect to x while treating y as a constant.

∫ [(x.y) . ((x^2 + y^2 + 25)^(3/2))] dx

To integrate this, we can apply the power rule by adding 1 to the exponent and dividing by the new exponent:

= (1/2) . (x^2 + y^2 + 25)^(5/2) / (5/2)

= (2/5) . (x^2 + y^2 + 25)^(5/2)

Now we have the result of the inner integration:

∫ [(x.y) . ((x^2 + y^2 + 25)^(3/2))] dx = (2/5) . (x^2 + y^2 + 25)^(5/2) + C

Now we will integrate this result with respect to y using the given limits of integration.

∫ [(2/5) . (x^2 + y^2 + 25)^(5/2) ] dy

To integrate this, we treat x as a constant and apply the power rule again:

= (2/5) . (x^2 + 25)^(5/2) . ∫ [(y^2 + 25)^(5/2)] dy

Now, we integrate the remaining function with respect to y:

= (2/5) . (x^2 + 25)^(5/2) . (2/7) . (y^2 + 25)^(7/2)

= (4/35) . (x^2 + 25)^(5/2) . (y^2 + 25)^(7/2) + C

Now we have the final result of the double integration:

∫ ∫ [(x.y) . ((x^2 + y^2 + 25)^(3/2))] dx dy = (4/35) . (x^2 + 25)^(5/2) . (y^2 + 25)^(7/2) + C

Please note that "C" represents the constant of integration, and it is customary to include it when performing indefinite integration.