# Elementary exponential integral

• Master J
In summary, the conversation discusses the integral of e^{i2t} dt between t and 0, with the use of u-substitution and complex numbers. There is also confusion about the integral being squared and the correct answer is given as 2(1 - coswt) / w.
Master J
I'm sure this integral is easy, but could someone perhaps show the working of:

$$\int$$ e$$^{i2t}$$ dt between t and 0.

I've tried it with trigonometric identities and keep getting lost!

Cheers!

Try a u-substitution.

u=2it
du=2i dt

Why do you need to do a substitution?

It is exactly the same as

$$\int_0^t e^{at}\,dt$$

except that $$a$$ is a complex number.

Although strictly speaking, it's bad form to use the same variable in your bounds and your integrand.

Cheers guys.

Yea true I guess, perhaps it should have been tau as the differential.

This relates to the integral of exp[ iwt], between t and 0. This is then squared, so i was going to just integrate exp [ 2iwt].

The answer is 2(1 - coswt) / w ...i can't seem to get this at all. Any ideas?

Is it the integral that's squared, or just the integrand?

The integral
$$\int_0^t e^{-2i\tau}d\tau$$
is
$$-\fra{1}{2i}e^{-2it}= \frac{i}{2}e^{-2it}$$

NOT
$$\frac{2(1- cos(\omega t))}{\omega}$$
or even
[tex]\frac{2(1- cos(2t))}{2}= 1- cos(2t)[/itex]

## 1. What is an elementary exponential integral?

An elementary exponential integral is a type of mathematical function that involves integrals of exponential functions. It is used to calculate the area under the curve of an exponential function.

## 2. What is the purpose of an elementary exponential integral?

The purpose of an elementary exponential integral is to solve problems involving exponential functions, such as population growth, radioactive decay, and compound interest.

## 3. How is an elementary exponential integral different from a regular integral?

An elementary exponential integral is a type of special function that can be expressed in terms of elementary functions (polynomials, trigonometric functions, exponential functions, and logarithmic functions). Regular integrals, on the other hand, may involve more complex functions that cannot be expressed in terms of elementary functions.

## 4. What are the common applications of elementary exponential integrals?

Elementary exponential integrals are commonly used in physics, chemistry, biology, economics, and other fields to model real-world phenomena that involve exponential growth or decay.

## 5. How can one solve an elementary exponential integral?

Solving an elementary exponential integral involves using integration techniques, such as substitution, integration by parts, or partial fraction decomposition. It may also require the use of tables or computer software to find the exact solution.

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