Second Fundamental Theorem of Calculus

Question

Find the value of f(x) = d/dx [Si(x^0.5)].

f(x) = _________

Answer 1

It looks like you actually want to apply part one of the Fundamental Theorem of Calculus (FTC) on the Si(x^0.5) function. Generally when you want to apply this part of the FTC, the upper limit of integration (the number or function on the top part of the integral sign) needs to have a linear term. In this case however you have a square root so you handle it slightly differently.

When trying to solve these types of problems you want to make a substitution for the upper limit of integration (assuming the upper limit is some function of x). In this case the substitution is u=x^0.5. From here you use the Chain Rule which will look like f(x)= d/du[Si(u)] • du/dx. This equation means f(x) equals the derivative of the integral of the Si function with respect to u (←use FTC part 1), times the derivative of u with respect to x (←Chain Rule). I hope this helps!