Question

Integration as the Inverse of Differentiation, Integration of ax" and Integration of the Functions of the Sum or Difference of Algebraic Terms

For each of the following, find f(x).
(a) f'(x) = x³ + 4x -1 1
(b) f'(x) = 1/2x² + root x
(c) f'(x) = 3x - 4 / 2x³
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Answer 1

Explanation:

The antiderivative for b is definitely incorrect, as stated in the comments section.

`\int\limits {(1)/(2x^2)+√(x) } \, dx \\(1)/(2) \int\limits{(1)/(x^2) } \, dx+\int\limits {x^{(1)/(2)} } \, dx` That's a bit more simplified. One more important simplification and then we can integrate one term at a time:

`(1)/(2)\int\limits{x^(-2)} \, dx +\int\limits{x^{(1)/(2) } \, dx` and here we go:

`(1)/(2)((x^(-2+1))/(-2+1))+(\frac{x^{(1)/(2)+(2)/(2)} }{(1)/(2)+(2)/(2) })+C` and

`(1)/(2)((x^(-1))/(-1))+(\frac{x^{(3)/(2)} }{(3)/(2) })+C` and finally,

`-(1)/(2x)+(2)/(3)x^{(3)/(2)}+C`

Answer 2

The answer is in the image above