The Fundamental Theorem of Calculus
Key Questions
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If we can find the antiderivative function
F(x) of the integrandf(x) , then the definite integralint_a^b f(x)dx can be determined byF(b)-F(a) provided thatf(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 takings(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.
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int_a^b f(x) dx=F(b)-F(a) ,
where F is an antiderivative off
Questions
Introduction to Integration
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Sigma Notation
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Integration: the Area Problem
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Formal Definition of the Definite Integral
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Definite and indefinite integrals
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Integrals of Polynomial functions
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Determining Basic Rates of Change Using Integrals
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Integrals of Trigonometric Functions
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Integrals of Exponential Functions
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Integrals of Rational Functions
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The Fundamental Theorem of Calculus
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Basic Properties of Definite Integrals