Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int sqrt(6 + r^3) dr` from `[7, x^2]`?

Answer 1

`d/dx int_7^(x^2)sqrt(6+r^3)dr=2xsqrt(6+x^6)`

Explanation:

Let `F(r) + C = intsqrt(6+r^3)dr`, that is, the function such that `d/(dr)F(r) = sqrt(6+r^3)`. The second fundamental theorem of calculus states that

`int_a^bsqrt(6+r^3)dr = F(b)-F(a)`

With that, we have

`d/dx int_7^(x^2)sqrt(6+r^3)dr = d/dx(F(x^2)-F(7))`

`=d/dxF(x^2) - d/dxF(7)`

`=sqrt(6+(x^2)^3)(d/dxx^2)-0`

`=2xsqrt(6+x^6)`

where in the second to last step we used the chain rule along with the fact that `F(7)` is a constant, and thus has a derivative of `0`.