Answer :

To find the most general antiderivative of the given function, you need to calculate the antiderivative of each term separately and combine them with a constant of integration. The antiderivative of 7x⁶ is x⁷, and for 6sec(x)tan(x) it is 3sec²(x). Finally, the most general antiderivative of f(x) is x⁷ + 3sec²(x) + C.

Find the most general antiderivative of f(x)=7x⁶+6sec(x)tan(x).

  1. Calculate the antiderivative of 7x⁶, which is (7/7)x⁷ = x⁷.
  2. For the antiderivative of 6sec(x)tan(x), use u-substitution: Let u = sec(x), then du = sec(x)tan(x)dx. So the antiderivative becomes 6u du = 3u² + C = 3sec²(x) + C.
  3. Therefore, the most general antiderivative of f(x) is x⁷ + 3sec²(x) + C, where C is the constant of integration.

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