How do you integrate #f(x)=(1+lnx)^4/x# using the quotient rule?

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Answer 1 · 2017-09-11T17:54:32

#int((1+lnx)^4)/xdx=1/5(1+lnx)^5+C#

There is no quotient rule for integration. Instead we will have to approach this another way.

#int((1+lnx)^4)/xdx#

let's see what happens when we differentiate #(1+lnx)^5#

#y=(1+lnx)^5#

#u=1+lnx=>(du)/(dx)=1/x#

#y=u^5=>(dy)/(du)=5u^4=5(1+lnx)^4#

#:.(dy)/(dx)=(dy)/(du)(du)/(dx)#

#(dy)/(dx)=5(1+lnx)^4xx1/x=(5(1+lnx)^4)/x#

we notice that comparing this with the integral it is same except for the constant, so we conclude:

#int((1+lnx)^4)/xdx=1/5(1+lnx)^5+C#