# How Can I Correctly Integrate e^(ix)cos(x)?

• flux!
In summary, integrating e^(i*x)cos(x) involves using the substitution method to replace the complex exponential function with a variable u. This results in an integral that can be solved using integration by parts and simplifying the trigonometric functions. The final answer will involve a combination of sine and cosine terms, and can be verified using Euler's formula.
flux!
I'am trying to prove

$$\int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix}$$

Wolfram tells so http://integrals.wolfram.com/index.jsp?expr=e^(i*x)cos(x)&random=false

But I am stuck in obtaining the first term:

My step typically involved integration by parts:

let $u=e^{ix}cos(x)$ and $dv=dx$

so:

$$du=-e^{ix}sin(x)dx+icos(x)e^{ix}$$$$du=ie^{ix}(sin(x)+cos(x))dx$$$$du=ie^{2ix}dx$$

the other one is just: $v=x$

then:
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)-i\int xe^{2ix}dx$$

We will again do another integration by parts for the second term, so we let $u=x$ and $dv=e^{2ix}dx$ then solving further, we obtain:
$$\int xe^{2ix}dx=\frac{-i}{2}xe^{2ix}+\frac{1}{4}e^{2ix}$$
plugging it back to the original problem, then doing simple distribution, we will obtain:
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)-i\left ( \frac{-i}{2}xe^{2ix}+\frac{1}{4}e^{2ix}\right )$$
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)- \frac{1}{2}xe^{2ix}-\frac{1}{4}ie^{2ix}$$
notice that we have proved the 2nd term, but the other half is badly away from what the Integration Table and Tools says:
$$\int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix}$$

What have I gone wrong?

flux! said:
I'am trying to prove

$$\int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix}$$

Wolfram tells so http://integrals.wolfram.com/index.jsp?expr=e^(i*x)cos(x)&random=false

But I am stuck in obtaining the first term:

My step typically involved integration by parts:

let $u=e^{ix}cos(x)$ and $dv=dx$

so:

$$du=-e^{ix}sin(x)dx+icos(x)e^{ix}$$$$du=ie^{ix}(sin(x)+cos(x))dx$$$$du=ie^{2ix}dx$$

the other one is just: $v=x$

then:
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)-i\int xe^{2ix}dx$$

We will again do another integration by parts for the second term, so we let $u=x$ and $dv=e^{2ix}dx$ then solving further, we obtain:
$$\int xe^{2ix}dx=\frac{-i}{2}xe^{2ix}+\frac{1}{4}e^{2ix}$$
plugging it back to the original problem, then doing simple distribution, we will obtain:
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)-i\left ( \frac{-i}{2}xe^{2ix}+\frac{1}{4}e^{2ix}\right )$$
$$\int e^{ix}cos(x) dx=xe^{ix}cos(x)- \frac{1}{2}xe^{2ix}-\frac{1}{4}ie^{2ix}$$
notice that we have proved the 2nd term, but the other half is badly away from what the Integration Table and Tools says:
$$\int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix}$$

What have I gone wrong?
You can side step a lot of the work above if you remember:

$$cos\, x = \frac{e^{ix}+e^{-ix}}{2}$$

flux!
I'm sorry, I couldn't follow your answer. But the integral is quite easy. Try using Euler's formula for e^x. Once you do ( and use a simple trigonometric identity) it's an easy
integration.
.
.
Edit: Or you can use the way in the post above mine, your convenience.

Last edited:
flux!
flux! said:
I'am trying to prove

$$\int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix}$$
All you really need to do is to show that ##\frac d{dx}[\frac x 2 - \frac 1 4 ie^{2ix}] = e^{ix}\cos(x)##. This isn't that hard to do.

flux!
Use streamking's approach $cos(x)=\frac{e^{ix}+e^{-ix}}{2}$. It is now a trivial problem.

flux!
Darn! Applied steamking's approached, solved in 2 lines only. -_-

Thanks everyone!

## 1. What is the meaning of "e^(i*x)cos(x)"?

"e^(i*x)cos(x)" is a mathematical expression that represents a complex number in the form of e raised to the power of an imaginary number multiplied by the cosine of a real number. This expression is often used in calculus, physics, and engineering to model oscillatory systems.

## 2. Why does "Stuck in Integrating e^(i*x)cos(x)" pose a challenge?

Integrating "e^(i*x)cos(x)" can be challenging because it involves both a complex number and a trigonometric function. The integration process requires using techniques such as substitution, integration by parts, or Euler's formula to simplify the expression and solve the integral.

## 3. What are some applications of "Stuck in Integrating e^(i*x)cos(x)"?

The expression "e^(i*x)cos(x)" is commonly used in physics and engineering to model oscillatory systems, such as electrical circuits, pendulums, and springs. It is also used in signal processing and control systems to analyze and design filters and control systems.

## 4. What are some tips for integrating "e^(i*x)cos(x)"?

When integrating "e^(i*x)cos(x)", it is helpful to first rewrite the expression using Euler's formula, which states that e^(i*x) = cos(x) + i*sin(x). This will help simplify the expression and make it easier to integrate. Additionally, choosing the appropriate integration technique, such as substitution or integration by parts, can also make the process easier.

## 5. Are there any other similar expressions to "e^(i*x)cos(x)"?

Yes, there are several similar expressions, such as e^(i*x)sin(x) and e^(i*x)tan(x). These expressions involve a complex number and a trigonometric function and are commonly used in physics and engineering to model oscillatory systems. The integration process for these expressions follows similar techniques as "e^(i*x)cos(x)".

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