How to Solve a Tricky Integration Using Prosthaphaeresis Formulas?

  • Thread starter yungman
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In summary, the conversation is about solving the integration of \int_0^h sin[\beta(h-z)] \; cos(\beta\;z \; cos \theta)\; dz using the prosthaphaeresis formula \sin\theta\cos\phi=\frac{\sin(\theta+\phi)+\sin(\theta -\phi)}{2}. The person asking for help had already tried a substitution method and a \cos(A+B) method, but neither were successful. After some time, they were able to figure out the solution using the prosthaphaeresis formula.
  • #1
yungman
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I need help how to solve this integration:

[tex]\int_0^h sin[\beta(h-z)] \; cos(\beta\;z \; cos \theta)\; dz[/tex]

I tried substitution of [itex]\;U=\beta(h-z)\;[/itex] and going nowhere. [itex]\; cos(A+B)\;[/itex] type of method won't work either because you can't get A and B out of this. Please help.

Thanks

Alan
 
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  • #2
yungman said:
I need help how to solve this integration:

[tex]\int_0^h sin[\beta(h-z)] \; cos(\beta\;z \; cos \theta)\; dz[/tex]

I tried substitution of [itex]\;U=\beta(h-z)\;[/itex] and going nowhere. [itex]\; cos(A+B)\;[/itex] type of method won't work either because you can't get A and B out of this. Please help.

Thanks

Alan

Use one of the prosthaphaeresis formulas:

[tex]\sin\theta\cos\phi=\frac{\sin(\theta+\phi)+\sin(\theta -\phi)}{2}[/tex]
 
  • #3
LCKurtz said:
Use one of the prosthaphaeresis formulas:

[tex]\sin\theta\cos\phi=\frac{\sin(\theta+\phi)+\sin(\theta -\phi)}{2}[/tex]

Sorry to acknowledge so late, took me a while to work it out.

Thanks
 

1. How do I determine the limits of integration?

The limits of integration are typically determined by the boundaries of the given function or region. You can also use a graph or table to help identify the limits.

2. What is the process for solving an integration problem?

The process for solving an integration problem involves first identifying the type of integration (definite or indefinite), then applying the appropriate integration techniques, such as substitution or integration by parts. The problem is then solved by using algebraic manipulations and basic rules of integration.

3. What are some common integration techniques?

Some common integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution. It is important to be familiar with each technique and know when to apply them in order to solve integration problems effectively.

4. How do I check my integration solution?

You can check your integration solution by taking the derivative of the integrated function and comparing it to the original function. If the two are equal, then your solution is correct.

5. What is the best way to practice and improve at solving integrals?

The best way to practice and improve at solving integrals is to work on a variety of problems, from basic to more complex, and to seek help and guidance from textbooks, online resources, or a tutor. It is also important to review and understand the different integration techniques and when to apply them.

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