Question

How can I find a particular antiderivative to satisfy particular conditions?
What I am given is: f(x)=(4x^-2)+(8x^-1)-3
and
F(1) must equal to 7. Any help appreciated :)

Answer 1

Note that you need to integrate term by term, i.e:\[\bf \int\limits_{ }^{}(4x^{-2}+8x^{-1}-3) \ dx=\int\limits_{}^{}4x^{-2} dx+\int\limits_{}^{}8x^{-1}dx+\int\limits_{}^{}-3dx\]Now one of the properties of integrals is that you can factor out any constants being multiplied, hence:\[\bf 4\int\limits\limits_{}^{}x^{-2} dx+8\int\limits\limits_{}^{}x^{-1}dx-3\int\limits\limits_{}^{}(1)dx \]Now note that you can re-write the '1' as 'x^0', like this:\[\bf 4\int\limits\limits_{}^{}x^{-2} dx+8\int\limits\limits_{}^{}x^{-1}dx-3\int\limits\limits_{}^{}x^0dx \]Now use the following and evaluate each integral separately:\[\bf \int\limits_{}^{}x^n \ dx=\frac{ x^{n+1} }{ n+1 }+C\]

Answer 2

@HazelL

Answer 3

Thank you very much :)

Answer 4

Don't forget the constant. Because ultimately what you'll end up doing is substituting x = 7 in to the integral that you get and you'll be equating that with 1. Then you'll have to re-arrange and solve for the constant 'C' to find EXACTLY what the integral is.

@HazelL

Answer 5

@genius12