The Fundamental Theorem of Calculus
Key Questions

If we can find the antiderivative function
#F(x)# of the integrand#f(x)# , then the definite integral#int_a^b f(x)dx# can be determined by#F(b)F(a)# provided that#f(x)# is continuous.We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that
#f(x)# is continuous and why.FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.
For most students, the proof does give any intuition of why this works or is true. But let's look at
#s(t)=int_a^b v(t)dt# . We know that integrating the velocity function gives us a position function. So taking#s(b)s(a)# results in a displacement. 
Fundamental Theorem of Calculus
#d/{dx}int_a^x f(t) dt=f(x)# This theorem illustrates that differentiation can undo what has been done to
#f# by integration.Let us now look at the posted question.
#f'(x)=d/{dx}\int_1^xsqrt{e^t+sint}dt=sqrt{e^x+sinx}# I hope that this was helpful.

#int_a^b f(x) dx=F(b)F(a)# ,
where F is an antiderivative of#f#
Questions
Introduction to Integration

Sigma Notation

Integration: the Area Problem

Formal Definition of the Definite Integral

Definite and indefinite integrals

Integrals of Polynomial functions

Determining Basic Rates of Change Using Integrals

Integrals of Trigonometric Functions

Integrals of Exponential Functions

Integrals of Rational Functions

The Fundamental Theorem of Calculus

Basic Properties of Definite Integrals