- #1
flyers
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Homework Statement
The Attempt at a Solution
I have no idea how to start this integral. Can anybody give me a hint to start it off?
Thank you.
Why Is the Definite Integral Zero for Symmetric Functions?
- Thread starter flyers
- Start date
In summary, the conversation is about how to start an integral and whether it is possible to solve it. The thread suggests sketching a graph and noticing the relationship between the parts for x<0 and x>0. It is concluded that the definite integral is zero due to the symmetry of the function.
I have no idea how to start this integral. Can anybody give me a hint to start it off?
Thank you.
- #2
Dick
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Sketch a graph. Anything strike you about the relation between the part of the graph for x<0 vs x>0?
- #3
flyers
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Well I noticed that x>0 is the reflection of x<0 except with negative values. Does this mean that the answer is 0? Also is it even possible to solve this integral?
- #4
Dick
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No, you aren't going to get far trying to find an indefinite integral. But the definite integral is zero. In general, if f(-x)=(-f(x)) then the integral over a symmetric interval around zero is zero. You can formally show this by splitting the integral into two parts and working out their relation by doing the substitution u=(-x).
1. How do I know which method to use to solve a definite integral?
The method used to solve a definite integral depends on the form of the integral. Some common methods include substitution, integration by parts, and trigonometric substitution. It is important to carefully analyze the integrand and choose the appropriate method.
2. What is the difference between a definite and indefinite integral?
A definite integral has specific limits of integration, while an indefinite integral does not. A definite integral gives a numerical value, while an indefinite integral gives a function. In other words, a definite integral represents the area under a curve within specific bounds, while an indefinite integral represents the general antiderivative of a function.
3. How do I evaluate a definite integral?
To evaluate a definite integral, you can use various integration techniques such as substitution, integration by parts, or trigonometric substitution. First, find the indefinite integral of the integrand, then plug in the limits of integration and subtract the result of the lower limit from the result of the upper limit.
4. Can definite integrals have negative values?
Yes, definite integrals can have negative values. The sign of a definite integral depends on the function being integrated and the limits of integration. It is possible for the area under a curve to be negative if the curve lies below the x-axis within the given bounds.
5. What are some common mistakes to avoid when solving a definite integral?
Some common mistakes when solving a definite integral include forgetting to add the constant of integration, integrating incorrectly, and using the wrong method or formula. It is important to carefully check your work and make sure all steps are correct.
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