Have functions fn(x)=ln^n(x) and F(x)=xlnx+ax+b. What are a,b if F is primitive pf f?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should consider n=1 such that:

`f_1(x) = ln x`

You need to remember that a primitive of a function is that function F(x) that differentiated yields f(x) such that:

`F'(x) = f(x)`

You need to differentiate F(x) with respect to x such that:

`F'(x) = x'*ln x + x*(ln x)' + (ax)' + b'`

`F'(x) = ln x + x*(1/x) + a`

`F'(x) = ln x + 1 + a`

Using the definition of primitive yields:

`F'(x) = f(x) =gt ln x + 1 + a = ln x =gt a + 1 = 0 =gt a = -1`

Notice that b may have any real value such that: `b in R`

Hence, evaluating a and b under given conditions yields `a = -1`  and `b in R` .