Solve Definite Integrals: Find F'(2)

[DollarBill]

Homework Statement

I'm not sure if I'm doing this right or not:

If F(x)=$$\int_0^x{\sqrt{t^3+1}dt}$$, then find F'(2)

The Attempt at a Solution

$$\int_0^x$$1/2(t³+1)^-1/2*3t²

1/2(x³+1)^-1/2*3x²

1/2(2³+1)^-1/2*3(2)²

Answer=2

 

[Knissp]

Think back to the (first) fundamental theorem of calculus.

By the way, your integration is wrong. What leads you to assume that

$$\int_0^x{\sqrt{t^3+1}dt}$$

is equal to
$$\int_0^x{1/2(t3+1)-1/2*3t2}$$ ??

It looks like you were thinking of a u-substitution, but that doesn't work.
Letting $$u = t^3+1$$, $$du=3t^2 dt$$, so $$dt = \frac{du}{3t^2}$$
You would then have to substitute this dt into your integral for a valid u-sub, but you notice that you would still have an expression in terms of t, so you cannot integrate with respect to u.

 

[DollarBill]

I was thinking of integration by reverse chain rule

 

[Dick]

Reverse chain rule is substitution. It doesn't work. As Knissp said, you can't integrate that function. You don't want the integral anyway, you want the derivative of the integral. Fundamental theorem of calculus.

 

[DollarBill]

Still not sure if I'm doing it right

$$\int_0^x{\sqrt{t^3+1}dt}$$

$${\sqrt{x^3+1}}$$

$${\sqrt{2^3+1}}$$-$${\sqrt{0^3+1}}$$

3-1=2

 

[w3390]

that is correct

 

[Dick]

Uh, not quite. Suppose F'(t)=sqrt(t^3+1), i.e. the antiderivative is F(t). Then the integral from 0 to x of sqrt(t^3-1)=F(x)-F(0), right? Find d/dx of that. The derivative of F(0) is zero, correct? It's a constant.

 

[Dick]

that is correct

It is not.

 

[Mark44]

Still not sure if I'm doing it right

$$\int_0^x{\sqrt{t^3+1}dt}$$

$${\sqrt{x^3+1}}$$

$${\sqrt{2^3+1}}$$-$${\sqrt{0^3+1}}$$

3-1=2

I concur with Dick.

Also, it would be helpful if you identified the things you're working with, rather than write a bunch of disconnected expressions.

$$F(x) = \int_0^x{\sqrt{t^3+1}dt}$$
$$So, F'(x) = \sqrt{x^3+1}$$
So far, so good. Now, what is F'(2)?

 

[DollarBill]

Ok, thanks got the answer (3). Yea, I probably should write out the expressions more often

 

[statdad]

Yea, I probably should write out the expressions more often

should read

Yea, I should write out the expressions more often.