To solve the given integral ∫(dx/2sin(x)3sec(x)), we can use trigonometric identities and integration techniques.

Step 1: Simplify the expression
Let's simplify the integrand using trigonometric identities:
sec(x) = 1/cos(x) and cosec(x) = 1/sin(x)
Therefore:
∫(dx/2sin(x)3sec(x)) = ∫(dx/2sin(x)3/cos(x))

Step 2: Rearrange the expression
To make the integration process easier, let's rearrange the expression by multiplying and dividing by necessary terms:
∫(dx/2sin(x)3/cos(x)) = ∫(dx/2sin(x)2/cos(x)) * (1/sin(x))

Step 3: Separate the terms
Now, we can separate the two terms in the integrand:
∫(dx/2sin(x)2/cos(x)) * (1/sin(x)) = ∫(1/2sin(x)2) * (1/cos(x)) * (1/sin(x)) dx

Step 4: Simplify the expression further
Let's simplify the expression by canceling out common terms:
∫(1/2sin(x)2) * (1/cos(x)) * (1/sin(x)) dx = ∫1/(2sin(x)cos(x)) dx

Step 5: Use a trigonometric identity
To proceed with the integration, we can use the identity: sin(2x) = 2sin(x)cos(x)
Substituting this identity into the integral:
∫1/(2sin(x)cos(x)) dx = ∫1/sin(2x) dx

Step 6: Apply a substitution
To solve this integral, we can use the substitution u = 2x:
Then, du = 2dx
Rearranging, dx = du/2

Substituting the value of dx and u into the integral:
∫1/sin(2x) dx = ∫1/sin(u) * (du/2) = (1/2) ∫cosec(u) du

Step 7: Integrate the expression
Using the integral of cosec(x), which is -ln|cosec(x) + cot(x)| + C, where C is the constant of integration, we can integrate the expression:
(1/2) ∫cosec(u) du = (1/2) * (-ln|cosec(u) + cot(u)|) + C

Step 8: Substitute back the value of u
Now, substitute back the value of u = 2x:
(1/2) * (-ln|cosec(2x) + cot(2x)|) + C

Step 9: Simplify the expression
The final answer to the integral is:
-(1/2) * ln|cosec(2x) + cot(2x)| + C

In conclusion, the integral of dx/2sin(x)3sec(x) is -(1/2) * ln|cosec(2x) + cot(2x)| + C, where C is the constant of integration.