The graph of f(x) and the x-axis bound three regions A B & C between x = -4 and x = 2 each of which has an area 3. Find the value of `int_(-4)^2 f(x)+2x+5 dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It is given that the graph of f(x) and the x-axis bound three regions A B and C between x = -4 and x = 2 each of which has an area 3. This is used while calculating the integral `int_(-4)^2 f(x) + 2x + 5 dx`

`int_(-4)^2 f(x) + 2x + 5 dx`

=> `3*3 - int_(-4)^(-2.5) 2x + 5 dx + int_(-2.5)^2 2x + 5 dx`

=> 9 - (x^2 + 5x) between -4 and -2.5 + (x^2 + 5x) between -2.5 and 2

=> 9 - ((-2.5)^2 - (-4)^2 + 5*(-2.5) - 5*(-4)) + (2^2 - (-2.5)^2 + 5*(2 + 2.5))

=> 9 - (6.25 - 16 - 12.5 + 20) + (4 - 6.25 + 22.5)

=> 9 + 2.25 + 20.25

=> 9 + 22.25

=> 31.25

The required area enclosed by the the given expression and the x-axis is 31.25.

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