# Evaluate the definite integral.  ∫-3^4(5-x/2)dx

Asked on by kimlee1983

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to evaluate the given definite integral such that:

`int_(-3)^4 (5-x)/2 dx`

You need to factor out `1/2`  such that:

`(1/2) int_(-3)^4 (5-x) dx`

You should use the property of linearity of integral, hence, you need to split the integral in two, such that:

`(1/2) int_(-3)^4 (5-x) dx = (1/2) int_(-3)^4 5 dx - (1/2) int_(-3)^4 x dx`

`(1/2) int_(-3)^4 (5-x) dx = (5/2) x|_(-3)^4 - (1/2)*(x^2/2)|_(-3)^4`

You need to use the fundamental theorem of calculus such that:

`(1/2) int_(-3)^4 (5-x) dx = (5/2) (4 - (-3)) - (1/2)((4^2)/2 - (-3)^2/2)`

`(1/2) int_(-3)^4 (5-x) dx = 35/2 - (1/2)(16/2 - 9/2)`

`(1/2) int_(-3)^4 (5-x) dx = 35/2 - (1/2)(7/2)`

`(1/2) int_(-3)^4 (5-x) dx = 35/2 - 7/4`

`(1/2) int_(-3)^4 (5-x) dx = (70 - 7)/4`

`(1/2) int_(-3)^4 (5-x) dx = 63/4 `

Hence, evaluating the given definite integral yields `int_(-3)^4 (5-x)/2 dx = 63/4.`

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