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?

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

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)

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

Just go with Steve's reply, ignore mine

I read that -3 as an extra term

To find the antiderivative of a function, you can use the power rule for integration and then add the constant of integration. Let's break down your steps to identify where the mistake may have occurred.

The given function is:
f(x) = x^5 - 4x^-3 - 4

To find the antiderivative F(x), we'll integrate each term separately.

1. For the term x^5, we can use the power rule for integration. Add 1 to the exponent and divide by the new exponent:
∫x^5 dx = (x^(5+1))/(5+1) = x^6/6

2. For the term -4x^-3, we integrate it by adding 1 to the exponent and dividing by the new exponent:
∫(-4x^-3) dx = (-4/(3+1))x^(1+1) = -x^2

3. The constant -4 can be integrated as -4x since the antiderivative of a constant is the constant multiplied by x.

Now we can sum up the individual antiderivatives and add the constant of integration, C:

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

Given that F(1) = 1, substitute x = 1 into the antiderivative to find the value of C:

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

Now, solve for C:

1 = -63/6 + C
1 = -21/2 + C
1 + 21/2 = C
23/2 = C

Therefore, the antiderivative F(x) that satisfies F(1) = 1 is:
F(x) = x^6/6 - x^2 - 4x + 23/2.