Differentiating under the integral

Thread starter ddcamp
Start date Oct 23, 2010
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Differentiating Integral

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SUMMARY

The discussion focuses on solving the integral from 0 to infinity of the function \(x^3 e^{-8x} \cos(5x)\) dx. Participants suggest using integration by parts as a method to tackle this problem. Additionally, the identity \(e^{iu} = \cos u + i \sin u\) is recommended to simplify the integration process. These techniques are essential for evaluating integrals involving exponential and trigonometric functions.

PREREQUISITES

Integration by parts
Complex exponential functions
Understanding of improper integrals
Basic knowledge of trigonometric identities

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Study the method of integration by parts in depth
Learn how to apply the identity \(e^{iu} = \cos u + i \sin u\) in integrals
Explore techniques for evaluating improper integrals
Investigate the use of Laplace transforms for similar integrals

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Students studying calculus, particularly those focusing on integral calculus, as well as educators and tutors looking for effective methods to teach integration techniques.

Oct 23, 2010

ddcamp

Homework Statement

integral 0 to infinite (x^3)(e^-8)(cos(5x)) dx

Homework Equations

integral by part?

The Attempt at a Solution

should I use integral by part?

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Oct 23, 2010

fzero

Science Advisor

Gold Member

Do you mean

[tex]\int_0^\infty x^3 e^{-8x} \cos(5x) dx?[/tex]

You might want to use the identity

[tex]e^{iu} = \cos u + i \sin u[/tex].

and then integrate by parts.