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Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use absolute values where appropriate.) f(x) = 1/5 − 3/x , x > 0

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MrsStrong

Answer:

[tex]\frac{1}{5}x - 3 ln x + C[/tex]

Step-by-step explanation:

The antiderivative of the function is the same as the integral. To find the integral or antiderivative, reverse the process of differentiation method on each term.

  • Currently the function 1/5 has degree 0. Through integration we increase the degree of each term by 1 and divide by the degree. So 1/5 becomes 1/5 x. But the degree of x is 1 so dividing by 1 only gives the same term.
  • For terms where x is in the denominator, we recognize it as a natural log function and write as ln x. Since -3/x is the original then it becomes - 3 ln x.

Thus 1/5 - 3/x becomes [tex]\frac{1}{5}x - 3 ln x + C[/tex]. We add a constant C for any possible constant that was eliminated during differentiation.

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