Showing that these two integrals are equal

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In summary: More generally, if ##f(x) = f(-x)## then ##\int_{-a}^a f(x) = 2\int_0^a f(x)##. On the other hand, if ##f(x) = -f(-x)## then ##\int_{-a}^a f(x) = 0##. Can you see why ##f(x) = e^x + e^{-x}## satisfies ##f(x) = f(-x)##?This should be almost self-evident and not difficult to argue from the symmetry around 0 between the two functions ##e^x## and ##e^{-x}##. If
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These two are equal to each other, but I can't figure out how they can be that.

1604049271124.png


I know that 2 can be taken out if its in the function, but where does the 2 come from here?
 
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  • #2
Why not evaluate the integrals to check they are the same?

If you do that you might see for yourself where the ##2## comes from.
 
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so do i solve for the integral on the left side and end up with the integral on the right?? Or do i solve them just as they are written above? (I have solved the equation with values, but don't really know how to do this one. )
 
  • #4
conv said:
so do i solve for the integral on the left side and end up with the integral on the right?? Or do i solve them just as they are written above?(I have solved the equation with values, but don't really know how to do this one. )
I suggest you evaluate both integrals. And check they are equal.

I assume you know how to integrate ##e^x## and ##e^{-x}##?
 
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  • #5
i understood it. Thanks
 
  • #6
conv said:
i understood it. Thanks

More generally, if ##f(x) = f(-x)## then ##\int_{-a}^a f(x) = 2\int_0^a f(x)##. On the other hand, if ##f(x) = -f(-x)## then ##\int_{-a}^a f(x) = 0##. Can you see why ##f(x) = e^x + e^{-x}## satisfies ##f(x) = f(-x)##?
 
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  • #7
This should be almost self-evident and not difficult to argue from the symmetry around 0 between the two functions ##e^x## and ##e^{-x}##. If in doubt about that, plot them. Which would only be reminding yourself of what you should securely know already. Not clever but rather basic.

ETA OK this is equivalent to what etotheipi says.
 

What is the purpose of showing that these two integrals are equal?

The purpose of showing that these two integrals are equal is to establish the equality of two mathematical expressions. This is important in order to simplify calculations and solve complex problems in mathematics and science.

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to calculate the total value of a function over a given interval.

What does it mean for two integrals to be equal?

When two integrals are equal, it means that the area under the curve of one function is the same as the area under the curve of another function over the same interval. This implies that the two functions have the same value over that interval.

How do you show that two integrals are equal?

To show that two integrals are equal, you must use mathematical techniques such as substitution, integration by parts, or trigonometric identities to manipulate one integral into the form of the other. This process is known as integration by substitution.

Why is it important to prove the equality of two integrals?

Proving the equality of two integrals is important because it allows us to solve complex mathematical problems and make accurate calculations. It also provides a deeper understanding of the relationship between different mathematical expressions and functions.

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