Consider the function f(x) = [ 5 /( x3 )] − [ 7 /( x 7 )].
Let F(x) be the antiderivative of f(x) with F(1) = 0.
Then F(x) =_____
Let F(x) be the antiderivative of f(x) with F(1) = 0.
Then F(x) =_____
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Take the antiderivative of f(x) (also known as an integral).
This can be split up, the integral of 5/x^3 + integral of -7/x^7
5/x^3 can be rewritten as 5*x^(-3). Use the power rule. The antiderivative is -15*x^(-4), or -15/x^4
7/x^7 is the same process. Rewrite it as -7x^(-7), and use the power rule. The antiderivative is 49*x^-8, or 49/x^7
Adding the solutions, we get F(x) = -15/x^4 + 49/x^8 + C
F(1)=0, so F(x)=0 when x = 1.
0=-15/(1)^4 + 49/(1)^8 + C
This becomes: 0 = -15 + 49 + C
Solving, C = -34
Finally, F(x) = -15/x^4 + 49/x^8 - 34
This can be split up, the integral of 5/x^3 + integral of -7/x^7
5/x^3 can be rewritten as 5*x^(-3). Use the power rule. The antiderivative is -15*x^(-4), or -15/x^4
7/x^7 is the same process. Rewrite it as -7x^(-7), and use the power rule. The antiderivative is 49*x^-8, or 49/x^7
Adding the solutions, we get F(x) = -15/x^4 + 49/x^8 + C
F(1)=0, so F(x)=0 when x = 1.
0=-15/(1)^4 + 49/(1)^8 + C
This becomes: 0 = -15 + 49 + C
Solving, C = -34
Finally, F(x) = -15/x^4 + 49/x^8 - 34