Fundamental Theorem of Calculus

[marknyoung]

I feel like I should be using the derivative in the [ ] areas instead of the sqrt(t^6+1) in the chain rule section?

 

[blamocur]

Are the variables $x$ and $t$ independent ? Also, why are you replacing $t$ with $x$ inside the square root?

 

[Steven G]

Can you state the fundamental theorem of calculus? You can't easily do this problem without know the theorem.

Also, why do you think that you just can change the order of the limits as you did?

 

[skeeter]

note …

$$\int_{x^3}^{10} \sqrt{t^6 +1} \, dt = -\int_{10}^{x^3} \sqrt{t^6+1} \, dt$$
also, if $a$ is a constant and $u$ is a function of $x$ …

$$\dfrac{d}{dx} \bigg[-\int_a^u f(t) \, dt \bigg] = -f(u) \cdot \dfrac{du}{dx}$$