# The rate of expenditure for maintenance of a particular machine is given by M'(x)=12x√x^2+5 where x is time measured in years.Total maintenance costs through the second year are $105. Find the...

The rate of expenditure for maintenance of a particular machine is given by

M'(x)=12x√x^2+5 where x is time measured in years.

Total maintenance costs through the second year are $105. Find the total maintenance function.

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You should find M(x), hence you need to integrate M'(x) such that:

`int M'(x) dx = int 12x*sqrt(x^2 + 5)dx`

You should come up with the substitution such that:

`x^2 + 5 = y =gt 2xdx = dy`

You need to write the equation `12x*sqrt(x^2 + 5)dx ` in terms of y such that:

`int 6*2x*sqrt(x^2 + 5)dx = 6*int sqrt y*dy`

`6*int sqrt y*dy = 6 int y^(1/2) dy`

`6 int y^(1/2) dy = 6*2/3*ysqrt y + c`

`int 6*2x*sqrt(x^2 + 5)dx = 4sqrt(x^2 + 5)^3 + c`

You need to substitute 2 for x in equation `M(x) = 4sqrt(x^2 + 5)^3 + c` such that:

`M(2) = 4sqrt(4 + 5)^3 + c =gt M(2) = 12*81 + c`

`M(2) = 972 + c =gt 105 = 972 + c =gt c = -867`

**Hence, evaluating total maintenance costs yields `M(x) = 4sqrt(x^2 + 5)^3 - 867.` **