Integration with Eulers formula.

It might be easier to do so if you make use of Euler's formula.In summary, the conversation discusses integrating the expression \frac{e^x}{cos(x)} using Euler's formula. The speaker suggests starting by writing \frac{e^x}{e^{ix}} and then integrating as usual to obtain the expression \frac{e^{(1-i)x}}{1-i}. They mention the usual method of multiplying and dividing by the complex conjugate and back substituting in e^(ix) to take the real part. Another speaker suggests an alternative method of back substituting in isin(x)+cos(x) on the bottom and then multiplying it by (1-i) to take the real part directly. This is not
  • #1
cragar
2,552
3
I want to integrate [itex] \frac{e^x}{cos(x)} [/itex] with eulers formula.
I start by writing [itex] \frac{e^x}{e^{ix}} [/itex]
then I integrate that as usual.
So after I integrate I get [itex] \frac{e^{(1-i)x}}{1-i} [/itex]
Normally I would multiply and divide by the complex conjugate and then back substitute in
e^(ix) and the take the real part.

can I just back substitute in isin(x)+cos(x) on the bottom and then multiply it by (1-i)
and then take the real part. That seems to easy.
Does anyone have suggestions. This is not a homework problem.
 
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  • #2
Hey cragar.

You should be able to integrate the expression and then take the real part as your answer.
 

What is Euler's formula?

Euler's formula is a mathematical equation that describes the relationship between exponential functions and trigonometric functions. It states that e^(ix) = cos(x) + isin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is any real number.

How is Euler's formula used in integration?

Euler's formula can be used to simplify the integration of complex exponential functions. By converting the exponential function into a trigonometric function, it becomes easier to integrate using standard integration techniques.

What is the benefit of using Euler's formula in integration?

Using Euler's formula can make integration of complex exponential functions more efficient and accurate. It can also help to solve integrals that may not be solvable using other methods.

Can Euler's formula be used in all integration problems?

No, Euler's formula is specifically useful for integrating complex exponential functions. It may not be applicable or helpful in other types of integration problems.

Are there any limitations or drawbacks to using Euler's formula in integration?

While Euler's formula can be a useful tool in integration, it may not always provide the most intuitive or straightforward solution. It also requires some understanding of complex numbers and trigonometric functions, which may be challenging for some individuals.

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