`f'(x) = 5x^4 - 3x^2 + 4, f(-1) = 2` Find `f`.

Textbook Question

Chapter 4, 4.9 - Problem 32 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate f and the problem provides f'(x), hence, you need to use the following relation, such that:

`int f'(x)dx = f(x)+ c`

`int (5x^4 - 3x^2 + 4)dx = f(x) + c`

You need to evaluate the indefinite integral of the power function, hence, you need to use the following formula:

`int x^(n) dx = (x^(n+1))/(n+1) + c`

`int 5x^4dx = 5(x^(4+1))/(4+1) + c => int 5x^4dx = x^5 + c`

`int 3x^2 dx =x^3 + c`

`int 4dx = 4x + c`

Hence,`f(x) = x^5 - x^3 + 4x + c` . You may find c using the following information, such that:

`f(-1) =2 => f(-1) = (-1)^5 - (-1)^3 + 4*(-1) + c => -1 + 1 - 4 + c = 2 => c = 6`

Hence, evaluating f(x) under the given condition, yields` f(x) = x^5 - x^3 + 4x + 6 .`

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