`f(x) = (1/2) + (3/4)x^2 - (4/5)x^3` Find the most general antiderivative of the function. (Check your answer by differentiation.)

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Chapter 4, 4.9 - Problem 3 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`f(x)=1/2+3/4x^2-4/5x^3 `

To determine the most general antiderivative of this function, take the integral of f(x).

`F(x)=intf(x) dx= int(1/2+3/4x^2-4/5x^3)dx`

Then, apply the integral formula `int cdu=cu + C` and `int u^ndu=u^(n+1)/(n+1)+C` .

`F(x) = 1/2x + 3/4*x^3/3-4/5*x^4/4+C=1/2x+x^3/4-x^4/5+C`

Therefore, the most general antiderivative of `f(x) = 1/2+3/4x^2-4/5x^3`  is `F(x)=1/2x+x^3/4-x^4/5+C` .

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