Integral: 𝛼^(2x)cos(x)dx Solution

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In summary, the problem involves finding the integral of e^2x*cos(x)dx, and the student has tried using u-substitution and integration by parts without success. They are seeking a hint or suggestion for solving the integral. Another student suggests integrating by parts twice, which eventually leads to a simplified form of the original integral. The student is grateful for the helpful suggestion.
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Homework Statement


[tex]\int[/tex][tex]e^{2x}cos(x)dx[/tex]

Homework Equations


The Attempt at a Solution


This isn't exactly a homework question, but it's close enough: it's from my old Calculus book (Larson), and I've been stuck on it for awhile. I just need a little hint or something. I've tried a u-substitution and haven't found anything that seems to help, and I've tried integration by parts, but nothing really simplifies--I just end up with an integral that is just as hard as before I try integration by parts. Maybe I'm using these techniques wrong or not seeing something obvious. No other 'harder' integration techniques should be needed (they come later in the book). Thanks!
 
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  • #2
Try integrating by parts twice. You'll eventually end up with a term that looks like the original integral. You can collect terms, so to speak, and solve for the integral.
 
  • #3
Ah! I would've never thought of that. Thanks!
 

FAQ: Integral: 𝛼^(2x)cos(x)dx Solution

What is the integral of 𝛼^(2x)cos(x)dx?

The integral of 𝛼^(2x)cos(x)dx is 1/4𝛼^(2x)sin(x) + C.

How do you solve the integral of 𝛼^(2x)cos(x)dx?

To solve the integral, you can use the substitution method where you let u = 𝛼^x and du = 𝛼^xln𝛼 dx. Then you can rewrite the integral as ∫u^2cos(ln(u))/ln𝛼 du. From there, you can use integration by parts to solve for the final result.

Can you simplify the integral of 𝛼^(2x)cos(x)dx?

Yes, you can simplify the integral by using the trigonometric identity cos(x) = (e^(ix) + e^(-ix))/2. This will result in an integral that is easier to solve.

Is there a specific value of 𝛼 that can make the integral of 𝛼^(2x)cos(x)dx equal to 0?

Yes, if 𝛼 = 0, then the integral of 𝛼^(2x)cos(x)dx will be equal to 0.

Can the integral of 𝛼^(2x)cos(x)dx be used to solve real-world problems?

Yes, the integral can be used in a variety of scientific fields such as physics, engineering, and economics to solve real-world problems that involve exponential and trigonometric functions.

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