Help evaluating Definite Integral

In summary, the student is trying to integrate a function from -1 to 1 using x^n = (n+1)x^(n+1), but gets stuck because the power of the variable i is not -1.
  • #1
Gameowner
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Homework Statement



Hi guys, need help evaluating this integral.

Integrate from -1 to 1

(t^i)(t^5-t^4) dt

Where i just an arbitrary variable, (it's actually an index number of a matrix)

Homework Equations





The Attempt at a Solution



So I thought I would begin by expanding the bracket to get

(t^(5+i))+(t^(4+i)) dt

then just evaluating it like you do

((t^(6+i))/(6+i))+((t^(5+i))/(5+i)) from -1 to 1

then just plugging the -1 and 1 into the t's, to get

((-2)^(6+i)/(6+i))+(2^(5+i)/(5+i))
 
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  • #2
Hmm, somehow the minus sign in the original problem turned into a plus when you expanded the expression.

I'm not sure how you got the -2 and 2 on the last part. Maybe you are adding the exponent bases together, but that doesn't work. 5^x + 5^x does not equal 10^x.

Integration using x^n = (n+1)x^(n+1) only works if n is not -1, otherwise the integral of (x^n)dx with n=-1 becomes ln(x). However, if this is the first semester of single variable calculus, then I suppose they want you to assume that the power is not -1 and just integrate with x^n = (n+1)x^(n+1).
 
  • #3
your attempt at the solution seems right to me.. as long as the variable i does not depend on t..
There are two errors (i) you have changed the minus sign from the original Integral to a + sign when you are actually evaluating it.
and when you plug the limits -1 and 1, you should get, ((-1)^(6+i))/(6+i)-(1/(6+i))-((-1)^(5+i))/(5+i)+(1/(5+i)).
you have plug the upper limit and the lower limit into each of the individual terms in the sum.

Hope this helps =)
 
  • #4
Nope, this is a fourth year optimization course. This question came out of approximating a function using a polynomial which lead to all this matrix index integral evaluation.

Always seem to get tricked up with the simple calculus ><

Yes, you guys are right, I don't know why I typed + instead of minus, I have here on my working minus, typo...

I also realized where I gone wrong, yea I added the exponent bases together which I shouldn't have done, if I take all those errors I made. I should get the answer that mathgeek4 posted.

Thanks for all your help!
 

1. What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve between two points on a graph. It is represented by the symbol ∫, and is calculated by finding the limit of a sum of infinitely many rectangles under the curve.

2. How do you evaluate a definite integral?

To evaluate a definite integral, you first need to determine the limits of integration, or the values between which you want to find the area under the curve. Then, you can use various integration techniques such as substitution, integration by parts, or trigonometric identities to solve the integral and find the numerical value of the area.

3. What are the applications of definite integrals?

Definite integrals have various applications in mathematics, physics, and engineering. They are used to calculate areas, volumes, and other physical quantities in real-world problems. They are also used to solve differential equations and model continuous systems.

4. What are some common mistakes when evaluating definite integrals?

One common mistake when evaluating definite integrals is forgetting to add the constant of integration when using integration techniques. Another mistake is incorrectly setting up the limits of integration or using the wrong integration method for a given function.

5. How can I check if my answer to a definite integral is correct?

You can check your answer by using the Fundamental Theorem of Calculus, which states that the definite integral of a function can be calculated by finding the antiderivative of the function and evaluating it at the limits of integration. You can also use a graphing calculator or software to visualize the area under the curve and compare it to your answer.

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