Integrals Problem

Question

 Consider the function f(x)=(9)/(x^(2))-(7)/(x^(7)) . Let  F(x) be the antiderivative of f(x) with F(1)=0.

Then F(x)=

Answer 1

The answer comes by rewriting f(x) as f(x) = 9x^(-2) - 7x^(-7). Taking the antiderivative of x^n is x^(n-1)/n-1 + C, where C is some constant.

Then, F(x) = 9x^(-1)/(-1) - 7x^(-6)/(-6) + C

F(x) = -9/x + 7/(6x^6) + C

Then we can find C using F(1) = 0.

F(1) = -9/1 + 7/6 + C = 0

C = 47/6

Final answer: F(x) = -9/x + 7/(6x^6) + 47/6

Answer 2

The antiderivative is (-9/x)+[7/6x⁶)]+C

Make the substitution for x=1 to evaluate C.