consider the function f(x) = 10/x^2 - 8/x^7, x > 0. let f(x) be the antiderivative of f(x) with f(1) = 0…

consider the function f(x) = 10/x^2 - 8/x^7, x > 0. let f(x) be the antiderivative of f(x) with f(1) = 0. then f(x) = video example: solving a similar problem

consider the function f(x) = 10/x^2 - 8/x^7, x > 0. let f(x) be the antiderivative of f(x) with f(1) = 0. then f(x) = video example: solving a similar problem

Answer

Explanation:

Step1: Recall antiderivative rules

The antiderivative of (x^n) is (\frac{x^{n + 1}}{n+1}+C) for (n\neq - 1). For (f(x)=\frac{10}{x^{2}}-\frac{8}{x^{7}}), rewrite it as (f(x)=10x^{-2}-8x^{-7}).

Step2: Find the antiderivative of each term

The antiderivative of (10x^{-2}) is (10\times\frac{x^{-2 + 1}}{-2+1}=- \frac{10}{x}), and the antiderivative of (-8x^{-7}) is (-8\times\frac{x^{-7 + 1}}{-7 + 1}=\frac{4}{3x^{6}}). So (F(x)=-\frac{10}{x}+\frac{4}{3x^{6}}+C).

Step3: Use the initial - condition

Given (F(1) = 0), substitute (x = 1) into (F(x)): (0=-10+\frac{4}{3}+C). Solve for (C): (C=10-\frac{4}{3}=\frac{30 - 4}{3}=\frac{26}{3}).

Step4: Write the final antiderivative

(F(x)=-\frac{10}{x}+\frac{4}{3x^{6}}+\frac{26}{3}).

Answer:

(F(x)=-\frac{10}{x}+\frac{4}{3x^{6}}+\frac{26}{3})