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Find general antiderivative of` f(x)=e^x + root (3)(x)` .

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kathyyyyyy123 | Student, Undergraduate | Honors

Posted December 19, 2012 at 8:04 AM via web

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Find general antiderivative of` f(x)=e^x + root (3)(x)` .

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Mary Joy Ripalda | High School Teacher | (Level 3) Educator

Posted December 19, 2012 at 9:37 AM (Answer #1)

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`f(x) = e^x + root(3)(x)`

Let  F(x) be the anti derivative of f(x).

To determine F(x), take the integral of f(x).

`F(x) = int f(x) dx = int (e^x + root(3)(x) ) dx = int e^xdx + int root(3)(x)dx`

To integrate the first term, apply the exponential formula `int e^u du = e^u + C` .

`= e^x + C + int root(3)(x) dx`

For the second term, express the radical as exponent. Then ,apply the power formula of integral which is `int u^n du = u^(n+1)/(n+1) + C ` .

`= e^x + C + int x^(1/3)dx`

`= e^x + C + x^ (4/3) /(4/3) + C`

`= e^x + C + (3x^(4/3))/ 4 + C`

Since C represents any number, we may re-write C + C as C only.

`= e^x + (3x^(4/3))/4 + C`

Hence, the anti derivative of `f(x) = e^x + roo(3)(x)` is `F(x)=e^x + (3x^(4/3))/4 + C` .

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