Integration by Parts for Complex Integrals

  • Thread starter donglepuss
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In summary, the integral of x^12 sinx dx is equal to x^12 -cosx - 12x^11 sinx - cosx (132x^11/11). To check this solution, one can differentiate the answer to see if it matches the original integrand. Another approach is to use recurrence relations, where I(n) = ∫x^n sin(x) dx and n is a positive integer. It is also possible to use integration by parts repeatedly, using the formula ∫udv = uv - ∫vdu.
  • #1
donglepuss
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Homework Statement
integral of x^12 sinx dx
Relevant Equations
answer: x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)
integral of x^12 sinx dx = x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)
 
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  • #2
donglepuss said:
Homework Statement: integral of x^12 sinx dx
Homework Equations: answer: x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)

integral of x^12 sinx dx = x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)
No, it's not.
However, you should get into the habit of checking your work when you have an integration problem. Just differentiate your answer, and if it is correct, you'll end up with the original integrand.
 
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  • #3
Might I suggest trying to use recurrence relations? I'm not sure if it's the best idea, though.

In any case, what I did was set ##I(n)=\int x^n \sin(x) \ dx## where ##n## is a positive integer.
 
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  • #4
It looks like you are trying to use integration by parts repeatedly, which should work if you do it right. As a reminder:
$$\int udv = uv - \int vdu$$
 
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Related to Integration by Parts for Complex Integrals

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is commonly used in calculus to calculate the total value of a function over a specific interval.

2. How do you know if an integral is correct?

An integral is considered correct if it follows the fundamental theorem of calculus, which states that the integral of a function is equal to the antiderivative of that function. This can be verified through various methods such as substitution, integration by parts, or using a table of integrals.

3. What are the common mistakes to avoid when solving integrals?

The most common mistake when solving integrals is forgetting to include the constant of integration. Other mistakes include mixing up the limits of integration, not properly simplifying the integral, and making algebraic errors.

4. How can I check my integral using technology?

There are many online calculators and software programs that can help you check your integral. These tools can solve the integral and show you the steps involved, allowing you to compare your own work and identify any mistakes.

5. Are there any tips for solving integrals more efficiently?

Some tips for solving integrals efficiently include memorizing common integrals, using substitution to simplify complex integrals, and practicing regularly. It is also helpful to review the fundamental properties of integrals, such as linearity and the power rule.

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