by Integr88 » Sat 02 April, 2011 16:53
The question say for the function ` f(x)= (2/x^2)-(6/x^7)` and let `F(x)=int(2/x^2)-(6/x^7)dx` so find initial equation
let's integrate `int(2/x^2)-(6/x^7)dx` (integrate means take the anti-derivatives
first rewrite the equation so it looks easy to take the antiderivatives
`int(2/x^2)-(6/x^7)dx=int(2x^(-2))-(6x^(-7))dx`
Now take the anti-derivatives (integrate):
`int(2x^(-2))-(6x^(-7))dx=2*x^(-2+1)/(-2+1)-6*x^(-7+1)/(-7+1)+C=2*x^(-1)/(-1)-6*x^(-6)/(-6)+C`
simplify and we have: `-2*x^(-1)+*x^(-6)+C=2x^(-1)+*x^(-6)+C`
`-2/x+1/x^6+C`
Now have F(1)=0 meaning when x=1 the whole equation is 0. so let's sub in the x=1
`0=-2/1+1/1^6+C`
`0=-2+1/1+C`
`0=-2+1+C`
`0=-1+C` adding 1 to both side and we have: `1=-1+1+C` we it becomes `1=C`
Now we have C=1, our equation is `F(x)=-2/x+1/x^6+1`