How Do You Integrate (e^ax)cos^2(2bx)dx Correctly?

In summary, the conversation is about integrating the expression (e^ax)cos^2(2bx)dx, where a and b are positive constants. The person has tried using Maple but it gave a ridiculous result, so they are asking for input on how to solve it. They have shown their progress so far, using the identity cosx = (e^ix - e^-ix)/2 and squaring inside the brackets, but they are still stuck and need to complete it by tomorrow.
  • #1
Geronimo85
20
0
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)*2[(e^(-4*b^2*x^2)-e^(4*b^2*x^2))/4]

I'm stuck and need to get this done for tomorrow
 
Last edited:
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  • #2
Why did you start a new thread for the same question? You are also still making the same mistake that was point out to you before:

Squaring inside the brackets does NOT give you

(e^ax)*2[(e^(-4*b^2*x^2)-e^(4*b^2*x^2))/4]

Because [itex](e^x)^2 = e^{2x}[/itex], NOT [itex]e^{x^2}[/itex].

And, although I shouldn't have to say it if you are taking calculus, (a+ b)2 is NOT a2+ b2!
 

Related to How Do You Integrate (e^ax)cos^2(2bx)dx Correctly?

1. What is an indefinite integral?

An indefinite integral, also known as an antiderivative, is a mathematical operation that is the inverse of differentiation. It involves finding a function that, when differentiated, will result in the original function.

2. How do you solve an indefinite integral?

To solve an indefinite integral, you need to use integration techniques such as u-substitution, integration by parts, or trigonometric substitution. First, you need to identify the function and its limits, then apply the appropriate integration method to find the antiderivative.

3. What is the difference between indefinite and definite integrals?

An indefinite integral yields a general solution, while a definite integral gives a specific numeric value. In other words, an indefinite integral results in a function, while a definite integral gives a single number as the answer.

4. When do you use integration to solve a problem?

Integration is used to solve problems in calculus, physics, engineering, and other fields that involve finding the area under a curve or the accumulation of a quantity over a given interval. It is also used to find the antiderivative of a function or to solve differential equations.

5. What are some real-life applications of indefinite integrals?

Indefinite integrals have various applications in real life, such as calculating the distance traveled by an object with a changing velocity, determining the amount of water in a tank with varying water levels, and finding the rate of change of a population over time. They are also used in economics, biology, and other fields to model and analyze real-world phenomena.

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