`f'(x) = sqrt(x^3) + root(3)(x^2)` Find `f`.

Textbook Question

Chapter 4, Review - Problem 67 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

Posted on

`f'(x)=sqrt(x^3)+root(3)(x^2)`

`f(x)=intf'(x)dx`

`f(x)=int(sqrt(x^3)+root(3)(x^2))dx`

apply the sum rule,

`f(x)=intsqrt(x^3)dx+introot(3)(x^2)dx`

`f(x)=x^(3/2+1)/(3/2+1)+x^(2/3+1)/(2/3+1)+C` , C is constant

`f(x)=x^(5/2)/(5/2)+x^(5/3)/(5/3)+C`

`f(x)=(2x^(5/2))/5+(3x^(5/3))/5+C`

 

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balajia | College Teacher | (Level 1) eNoter

Posted on

Given `f'(x) = sqrt(x^3)+root(3)(x^2)`

Integrating both sides with respect to x, we get

`intf'(x) = int(sqrt(x^3)+root(3)(x^2))dx`

`f(x)= int (x^(3/2)+x^(2/3))dx`

`=(2/5)x^(5/2)+(3/5)x^(5/3)+c`

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