Integration by Parts: Solving \int64x^2cos(4x)dx

In summary, the conversation discusses integrating the expression \int64x^2cos(4x)dx by parts. The individual asking the question initially tried this approach multiple times, but ended up with a complicated solution that was not correct. Another individual suggests using a substitution to simplify the expression before integrating by parts twice. The original individual then realizes they have already solved the problem and thanks the other person for their help.
  • #1
chrono210
6
0
How would you go about doing this:

[tex]\int64x^2cos(4x)dx[/tex]

The question specifically asks to integrate it by parts, so I integrated it that way a couple of times and came out with some long mess of sines and cosines, but it's not the right answer.

Thanks.
 
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  • #2
The approach is correct. The first thing I'd do is a substitution 4x->u just so that the numbers disappear. Then just integrate by parts twice. I suggest you try and post your attempt, so that we can see if it's just a simple algebraic mistake. Like I said, it's going in the right direction.
 
  • #3
Oops, I almost forgot about this. I actually was able to figure it out. Thanks though. :)
 

FAQ: Integration by Parts: Solving \int64x^2cos(4x)dx

What is integration by parts and how does it work?

Integration by parts is a method used in calculus to solve integrals of products of functions. It involves breaking down the integral into two parts and using the product rule of derivatives to solve for one part while leaving the other as a new integral. This process is repeated until the integral can be easily solved.

How do I know when to use integration by parts?

You should use integration by parts when the integrand (the expression being integrated) is a product of two functions, and one of the functions becomes simpler when differentiated while the other becomes simpler when integrated.

What are the steps for solving an integral using integration by parts?

The steps for integration by parts are:

  1. Identify the two functions in the integrand, and label one as u and the other as dv.
  2. Use the product rule to find the derivative of u and the antiderivative of dv.
  3. Plug these values into the integration by parts formula: ∫u*dv = uv - ∫v*du
  4. Solve for the remaining integral on the right side by repeating the process.
  5. If necessary, use algebraic manipulation to solve for the original integral.

How do I solve \int64x^2cos(4x)dx using integration by parts?

To solve this integral using integration by parts, you would start by labeling u = 64x^2 and dv = cos(4x)dx. Then, you would find du and v by taking the derivatives and antiderivatives of u and dv respectively. Plugging these values into the integration by parts formula, you would get uv - ∫v*du, which can then be solved using the same method. The final solution should be in terms of x and should be easily evaluated.

Are there any tips or tricks for using integration by parts?

Yes, some useful tips for using integration by parts include:

  • Choose u and dv strategically to make the integration process easier.
  • If the integral becomes more complicated after each iteration, try using integration by parts again on the remaining integral.
  • When using integration by parts multiple times, make sure to keep track of the u and dv labels to avoid confusion.
  • Always check your final answer by differentiating it to see if it matches the original integrand.

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