Question

Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f(x) = 8x3/5 + 3x−4/5

Answer 1

`(8x^(4))/(20)+(3x^(2))/(2)-(4)/(5)x+C`

Explanation:

Given the function:
`f(x)=(8x^3)/(5)+3x-(4)/(5)`

To take the antiderivative (or integral) of a function, we follow the format below.

`f(x)=x^n\\$Then its antiderivative\\Antiderivative of f(x)$=(x^(n+1))/(n+1)`

Therefore, the antiderivative of f(x) is:

`=(8x^(3+1))/(5(3+1))+(3x^(1+1))/(2)-(4)/(5)x+C\\=(8x^(4))/(20)+(3x^(2))/(2)-(4)/(5)x+C`

We want to check our result by differentiation.

`(d)/(dx)\left((8x^(4))/(20)+(3x^(2))/(2)-(4)/(5)x+C\right)\\=(d)/(dx)\left((8x^(4))/(20)\right)+(d)/(dx)\left((3x^(2))/(2)\right)-(d)/(dx)\left((4)/(5)x\right)+(d)/(dx)\left(C\right)\\\\=(32x^(3))/(20)+(6x)/(2)-(4)/(5)+0\\\\=(8x^(3))/(5)+3x-(4)/(5)`