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]#?

1 Answer
Oct 19, 2016

Answer:

#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#.