How to use Fundamental Theorem of Calculus. to solve this?

Question

Find the derivative of the following function
#F(x)=∫_sqrtx^1 (s^2)/(5+4s^4)ds#
using the appropriate form of the Fundamental Theorem of Calculus.
#F'(x)=?#

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Answer 1 · 2017-12-02T10:20:58

#F'(x)= -1/(2)*(sqrt(x))/(5+4(x^2)#

First rewrite so the upper bound is the function:

#-\int_1^sqrt(x)s^2/(5+4s^4)ds#

(Flip the bounds, flip the sign.)

Now use the 2nd FTC:

#d/(dx)(-\int_1^sqrt(x)s^2/(5+4s^4)ds)=-((sqrt(x))^2)/(5+4(sqrt(x))^4)*1/(2sqrt(x))#

Simplifying gives us:
#F'(x)= -1/(2)*(sqrt(x))/(5+4(x^2)#