Determine the definite integral of any polynomial.

Primary tabs

Lesson Parent: 
The Fundamental Theorem of Integral Calculus
Problem: 

Determine the definite integral
∫baPn(x)
where Pn(x)=cnxn+cn−1xn−1+⋯+c0.

Answer: 

∫baPn(x)=cnn+1{bn+1−an+1}+cn−1n{bn−an}+⋯+c0{b−a}

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

The function

<Sign in to see all the formulas>


is any arbitrary polynomial function. We know that we can find an antiderivative of a polynomial and it is the function

<Sign in to see all the formulas>


where C is any constant number.

Now we know that polynomial is a continuous function and is therefore integrable.

From the Fundamental Theorem of Integral Calculus we know that if <Sign in to see all the formulas> is integrable and that if F(x) is an antiderivative of <Sign in to see all the formulas> then

<Sign in to see all the formulas>


Therefore,

<Sign in to see all the formulas>

The proof is now complete.