What is the value of #F'(x)# if #F(x) = int_0^sinxsqrt(t)dt# ? Calculus Applications of Definite Integrals Solving Separable Differential Equations 1 Answer Ratnaker Mehta Jan 21, 2017 #:. F'(x)=(sqrtsinx)(cosx).# Explanation: #F(x)=int_0^sinx sqrttdt# # because, intsqrttdt=intt^(1/2)dt=t^(1/2+1)/(1/2+1)=2/3t^(3/2)+c,# #:. F(x)=[2/3t^(3/2)]_0^sinx# #:. F(x)=2/3sin^(3/2)x# #:. F'(x)=2/3[{(sinx)}^(3/2)]'# Using the Chain Rule, #F'(x)=2/3[3/2(sinx)^(3/2-1)]d/dx(sinx)# #=(sinx)^(1/2)(cosx)# #:. F'(x)=(sqrtsinx)(cosx).# Enjoy Maths.! Answer link Related questions How do you solve separable differential equations? How do you solve separable first-order differential equations? How do you solve separable differential equations with initial conditions? What are separable differential equations? How do you solve the differential equation #dy/dx=6y^2x#, where #y(1)=1/25# ? How do you solve the differential equation #y'=e^(-y)(2x-4)#, where #y5)=0# ? How do you solve the differential equation #(dy)/dx=e^(y-x)sec(y)(1+x^2)#, where #y(0)=0# ? How do I solve the equation #dy/dt = 2y - 10#? Given the general solution to #t^2y'' - 4ty' + 4y = 0# is #y= c_1t + c_2t^4#, how do I solve the... How do I solve the differential equation #xy'-y=3xy, y_1=0#? See all questions in Solving Separable Differential Equations Impact of this question 1561 views around the world You can reuse this answer Creative Commons License