# Integral of sin(e^x): Solve It Here

• ehj
In summary, the function is continuous, but it is not possible to find the integral using the method of integration by parts.
ehj
The function is continuous so the integral exists, but how do you find it :)?

It is highly unlikely that a closed form expression in terms of elementary functions exists, however, if you only wish to evaluate it as a definite integral then there are numerous methods of doing that.

$$\int sin(e^t) dt$$

Let $x=e^t \Rightarrow \frac{dx}{dt}=e^t=x$

$$\int sin(e^t) dt \equiv \int \frac{sinx}{x}dx$$

and that doesn't exist in terms of elementary functions

Hey!

Why can't you use the method of intergation by Parts for $$\int sin(e^x) dt$$?

We CAN think that this integral is in the form $$\int f(x) g(x) dx$$, right?

I think it can be done using the formula of integration by parts, $$\int uv' = uv - \int v u'$$. This might be done that way imo.

Last edited:
roam said:
Hey!
Why can't you use the method of intergation by Parts for $$\int sin(e^x) dt$$?

Go ahead and try. Be prepared to be frustrated, however because ...

rock.freak667 said:
$$\int sin(e^t) dt \equiv \int \frac{sinx}{x}dx$$
and that doesn't exist in terms of elementary functions

This latter integral is encountered in math and science quite frequently, so frequently that it has been given a name -- the sine integral. For more info, see

http://planetmath.org/encyclopedia/SinusIntegralis.html"
http://mathworld.wolfram.com/SineIntegral.html"
http://en.wikipedia.org/wiki/Sine_integral"

Last edited by a moderator:
roam said:
Hey!

Why can't you use the method of intergation by Parts for $$\int sin(e^x) dt$$?

We CAN think that this integral is in the form $$\int f(x) g(x) dx$$, right?
Why would you think so? If it were exsin(x), f and g would be obvious but here the only functions "multiplied" together are 1 and sin(ex). If you take u= 1 and dv= sin(ex)dx, you are back to the original problem. If you take u= sin(ex) and dv= dx you get du= cos(ex)exdx and v= x so you have gone from $\int udv= \int sin(e^x)dx$ to $uv- \int v du= xsin(e^x)- \int x cos(e^x)e^x dx$ which doesn't look any easier to me.

Well if you keep on doing integration by parts you get an infinitely long result, don't think that's of any use
It's just weird.. because my math teacher says that if the funciton is continuous the integral exists, but that must only be with a definite integral then.

I think we all agree the integral exists. Nobody here has said it doesn't.

What was said was:
rock.freak667 said:
... that doesn't exist in terms of elementary functions

The integral exists, but we can't express it in terms of elementary functions.

Just because some function cannot be expressed as a sum of a finite number of combinations of elementary functions does not mean the function does not exist. It just means that the function in question is not an elementary function, and that is all it means. The function can still be expressed as an infinite series, for example.

## 1. What is the integral of sin(e^x)?

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.

## 2. How do I solve the integral of sin(e^x)?

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.

## 3. Can the integral of sin(e^x) be solved using any other method?

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.

## 4. What is the domain and range of the integral of sin(e^x)?

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.

## 5. Can the integral of sin(e^x) be used in real-world applications?

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.

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