# Integrating Simple Expression: Solving (e^ax)cos^2(2bx)

• Geronimo85
In summary, the conversation is about integrating the expression (e^ax)cos^2(2bx)dx where a and b are positive constants. The person is struggling with finding a simple way to integrate it and is seeking input from others. They have tried using the trigonometric identity for cos^2(2bx) but are still having trouble. Another person suggests using cos^2(2bx)=[1+cos(4bx)]/2 and points out that the final integral will not look pretty. The original poster apologizes for reposting the question multiple times and expresses frustration with not being able to solve it.
Geronimo85
I'm supposed to integrate the following expression, and supposedly there is a very simple way to do so. Maple comes up with something rediculous, so I'd appreciate any input. Sorry about the short hand, don't know how to make everything pretty on here:

Integral[(e^ax)cos^2(2bx)dx] where a and b are positive constants

So far all I've got is:

(e^ax)cos^2(2bx)= (e^ax)*[(e^(i*2*b*x) - e^(-i*2*b*x))/2]^2

because: cosx = (e^ix - e^-ix)/2

squaring inside the brackets gets me:

(e^ax)* [((e^(i*2*b*x)-e^(-i*2*b*x)/2)^2]

I'm just really not getting something here

$$e^{ax}.e^{-bx} = e^{(a-b)x}$$

You should be able to work from that, just factor the exponents. The final integral won't look pretty.

How many times are you going to post this same question?

Use cos^2(2bx)=[1+cos(4bx)]/2.

sorry about reposting it, I just feel like I'm going in circles

## 1. What is the purpose of integrating simple expressions?

Integrating simple expressions allows us to find the area under a curve, which is useful in many areas of science such as physics and engineering.

## 2. How do you solve (e^ax)cos^2(2bx)?

To solve this expression, we can use the product rule of integration. First, we integrate e^ax using the formula 1/a * e^ax + C. Then, we integrate cos^2(2bx) using the formula 1/2 * (x + 1/4sin(4bx)) + C. Finally, we multiply the two results together to get the final answer.

## 3. Can we use substitution to solve this expression?

Yes, we can use substitution to solve this expression. We can substitute u = 2bx and du = 2b dx, which simplifies the expression to e^(au/2b)cos^2(u).

## 4. Is it possible to solve this expression without using integration?

No, integration is the only method for solving expressions like this. However, we can use numerical methods such as Simpson's rule or the trapezoidal rule to approximate the solution if needed.

## 5. How can we apply this concept in real-life situations?

In science, we often encounter situations where we need to find the area under a curve or the total amount of a substance over time. Integration of simple expressions allows us to solve these problems and make predictions in fields such as chemistry, biology, and economics.

Replies
4
Views
803
Replies
3
Views
1K
Replies
2
Views
6K
Replies
5
Views
1K
Replies
4
Views
1K
Replies
2
Views
935
Replies
1
Views
1K
Replies
12
Views
2K