Question

For the function f, given below, find the antiderivative F that satisfies F(1)=1.

f(x)= x^5 -4x^-3 -4 .

I keep getting x^6/6 +x^2 -4x + 17/6 . But my answer is wrong


My other answer of x^6/6 +4x^-2 +4x + 43/6 is wrong also. They both end up equaling 1. So what am I doing wrong?

Answer 1

If you differentiate your answers, you certainly don't get back the original f(x), so you are clearly wrong
your problem is the middle term -4x^-3

for simple single terms like this, here is how I go about it to integrate
I write down the given term: -4x^-3
I then raise the exponent by 1 ---> -4x^-2
I then divide by the new exponent ---> -4x^-2 /-2

= 2x^-2
---- differentiate that and you will get -4x^-3

I also noticed that you probably have a typo near the end
f(x) = ..... -3 - 4
that second-last term is probably -3x

so F(x) = (1/6)x^6 + 2x^-2 - (3/2)x^2 - 4x + c
but (1,1) is supposed to satisfy this
1 = (1/6) + 2 + 3/2 - 4 + c
1 = 1/3 + c
c = 2/3

Check by taking the derivative of F(x)

Answer 2

Close, but no cigar. For powers, raise it by one and divide by it. So,

F(x) = 1/6 x^6 -4(1/-2) x^-2 - 4x + c
= 1/6 x^6 + 2x^-2 - 4x + c

Since F(1) = 1,

1 = 1/6 + 2 - 4 + c
c = 17/6

F(x) = 1/6 x^6 + 2x^-2 - 4x + 17/6

Frankly, I think the negative exponent is a typo, but that's for you to check. If it is, then

F(x) = x^6/6 - x^4 - 4x + c
and
1 = 1/6 - 1 - 4 + c
c = 23/6

Answer 3

Just go with Steve's reply, ignore mine

I read that -3 as an extra term

Answer 4

Thank You Steve. Also Reiny that's what you get for being rude. >:(