The work is calculated by replacing pressure with nRT/V then pulling nRT outside the integral. We are only allowed to pull T outside the integral in that example because: a. we are always allowed to pull T outside a volume integral
b. T is constant for an isothermal process
c. it's an expansion not a compression
d. T has a positive value in that problem

b. T is constant for an isothermal process

Explanation:

In a general case for an ideal gas, we have the relation:

dW = -pdV.

If we integrate in both sides, we get:

W = ∫-pdV

Now, as the problem says, we can replace p by n*R*T/V

This is because of the equation for ideal gases.

p = n*R*T/V

replacing that in the equation we get

W = -∫pdV = -∫(n*R*T/V)dV

Now we want to take the n*R*T part outside of the integral.

But we only could do this if T does not depend on V.

If T is cosntant, then T does not depend on V, and we know that T is constant when we are in an isothermal process.

Where an isothermal process is a process where the temperature does not change, then T = constant.

Then n*R*T = constant = k

Because this is a constant, we could take it out of the integral so we get:

W = -∫(n*R*T/V)dV ) = -(n*R*T)∫(1/V)*dV

Then the correct option is option b.

b. T is constant for an isothermal process

d is the correct answer

the answer would be a