Use PFD(partical fraction decomposition) to integrate: {x^3+1/x(x^2+4)^2 dx

Firm's production is represented by z=(10(x^(1/2)))*y^(1/2). z=amount of output. x=amount of labor used. y=amount of capital used. Labor cost= 20 dollars per unit, Capital cost= 80 dollars per unit. Part a) 100 units are currently being made as the output. Labor's cost minimizing quantity=20. Capital's cost minimizing quantity=5. Draw a graph of this with both isocosts and isoquants. Part b) The business wants to make the output increase from 100 units to 140 units. In the short run, capital is fixed. What is the amount of labor needed? Show this in your graph in part a. Calculate the new total cost. Part c) In the graph from part a, draw and label a point that shows capital's and labor's cost minimizing level in the long run if the business wants to make 140 units.

Can you please explain to me how you went from x^4+1 dydx to x^4+1 (x^3)?

For this second step after we have (a^-1ba)^k(a^-1 ba)^1, shouldn't we have a^k-1 b^k a^k a^-1 b^1 a^1?