Solution:

Given differential equation:

(dy/dx) = [(x(2logx + 1))/(sin(y) * y * cos(y))]

To solve this differential equation, we can use the method of separation of variables.

1. Separate the variables:
(dy)/(sin(y) * y * cos(y)) = (x(2logx + 1))/dx

2. Integrate both sides with respect to their respective variables:

∫(dy)/(sin(y) * y * cos(y)) = ∫(x(2logx + 1))/dx

3. Evaluate the integrals:

To evaluate the integral on the left side, we can use a substitution. Let u = sin(y), then du = cos(y)dy. The integral becomes:

∫(1/(u * y)) du = ln|u| + C1

To evaluate the integral on the right side, we can use the power rule of integration. The integral becomes:

∫(x(2logx + 1))/dx = ∫(2xlogx + x)dx = x^2logx + (1/2)x^2 + C2

4. Combine the integrals:

ln|sin(y)| + C1 = x^2logx + (1/2)x^2 + C2

5. Combine the constants:

Let C = C2 - C1. The equation becomes:

ln|sin(y)| = x^2logx + (1/2)x^2 + C

6. Exponentiate both sides:

|sin(y)| = e^(x^2logx + (1/2)x^2 + C)

Since sin(y) can be positive or negative, we remove the absolute value signs and obtain:

sin(y) = ±e^(x^2logx + (1/2)x^2 + C)

7. Simplify the expression:

Using properties of exponents, we can rewrite the right side as:

sin(y) = ±e^C * e^(x^2logx) * e^((1/2)x^2)

Let K = ±e^C, then:

sin(y) = K * e^(x^2logx) * e^((1/2)x^2)

8. Solve for y:

Taking the natural logarithm of both sides, we have:

ln(sin(y)) = ln(K * e^(x^2logx) * e^((1/2)x^2))

Using properties of logarithms, we can simplify this to:

ln(sin(y)) = ln(K) + x^2logx + (1/2)x^2

9. Exponentiate both sides:

sin(y) = e^(ln(K) + x^2logx + (1/2)x^2)

Using properties of exponents, we have:

sin(y) = Ke^(x^2logx) * e^((1/2)x^2)

Therefore, the solution to the given differential equation is:

ysin(y) = Ke^(x^2logx) * e^((1/2)x^2), where K is a constant.

The correct answer is option 'D': ysiny = x^2logx.