Use the substitution u=4x-1 to find the exact value of 1/4<int<1/2 ((5-2x)(4x-1)^1/3)dx

We are required to solve this integral using integration by substitution, in which we assign a variable to equal a certain region of the integrated function in this case, 4x-1. The purpose of this is to remove of the remaining integral, by changing the derivative such that the function is integratable. so if u=4x-1 then du/dx=4, and thus dx=du/4, now by substitution, int(5-2x)(4x-1)^1/3dx= int(5-(u+1)/2)/4(u^1/3)du; in this instance x=(u+1)/4 therefore 5-2x=5-(u+1)/2. Now by expanding the brackets we have int((5/4)-(u/8)-(1/8))(u^1/3)du=int((5u^1/3/4)-(u^4/3/8)-(u^1/3/8))du=int(9u^1/3/8)-(u^4/3/8)du. Now this integral is solvable, & so = [(27u^4/3/32)-3u^7/3/56]; what's more the limits of this integral will change when the subtitution is carried out. Simply sub, 1/2&1/4 into 4x-1, and they become 1 and 0, therefore the value of the integral is 27/32-3/56-0= 177/224