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ehj
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The function is continuous so the integral exists, but how do you find it :)?
roam said:Hey!
Why can't you use the method of intergation by Parts for [tex]\int sin(e^x) dt[/tex]?
rock.freak667 said:[tex]\int sin(e^t) dt \equiv \int \frac{sinx}{x}dx[/tex]
and that doesn't exist in terms of elementary functions
Why would you think so? If it were e^{x}sin(x), f and g would be obvious but here the only functions "multiplied" together are 1 and sin(e^{x}). If you take u= 1 and dv= sin(e^{x})dx, you are back to the original problem. If you take u= sin(e^{x}) and dv= dx you get du= cos(e^{x})e^{x}dx and v= x so you have gone from [itex]\int udv= \int sin(e^x)dx[/itex] to [itex]uv- \int v du= xsin(e^x)- \int x cos(e^x)e^x dx[/itex] which doesn't look any easier to me.roam said:Hey!
Why can't you use the method of intergation by Parts for [tex]\int sin(e^x) dt[/tex]?
We CAN think that this integral is in the form [tex]\int f(x) g(x) dx[/tex], right?
rock.freak667 said:... that doesn't exist in terms of elementary functions
The integral of sin(e^x) is an indefinite integral, which means it does not have a specific numerical answer. It can be written as ∫sin(e^x)dx.
One way to solve the integral is to use the substitution method by letting u = e^x. This will transform the integral into ∫sin(u)du, which can then be solved using integration by parts or trigonometric identities.
Yes, there are other methods such as using the power series expansion of sin(x) or using the complex exponential function. However, these methods may be more complex and may not be suitable for all situations.
The domain of the integral is all real numbers, as it can be solved for any value of x. The range of the integral depends on the limits of integration and the value of x, but it will always be a continuous function between -1 and 1.
Yes, the integral of sin(e^x) can be used in various fields such as physics, engineering, and economics to model real-world phenomena involving oscillations and periodic functions. It can also be used for signal processing and data analysis.