How to Integrate Complicated Volume Integrals of Spheres?

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SUMMARY

This discussion focuses on integrating complicated volume integrals of spheres, specifically addressing the integral of (sinx)**3 and the method of integrating sinx(cosx)**2. The conversation highlights the substitution technique, where the integral can be transformed into a simpler form using the relationship a**u du, with du representing sinx * dx. The participants emphasize the importance of recognizing patterns in trigonometric integrals to simplify calculations.

PREREQUISITES
  • Understanding of trigonometric integrals, specifically (sinx)**3 and sinx(cosx)**2.
  • Familiarity with integration techniques, including substitution methods.
  • Knowledge of volume integrals in spherical coordinates.
  • Basic calculus concepts, particularly differentiation and integration.
NEXT STEPS
  • Research the integral of (sinx)**3 using integration by parts.
  • Study substitution methods in trigonometric integrals for simplification.
  • Explore volume integrals in spherical coordinates for complex shapes.
  • Learn about advanced integration techniques, such as contour integration.
USEFUL FOR

Mathematicians, physics students, and anyone involved in advanced calculus or integration techniques, particularly those dealing with trigonometric functions and volume calculations.

geft
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I can't seem to get the answer. How to integrate when it's this complicated?
 

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Assuming correctness up to the next to last line, you can look up the integral of (sinx)**3. The integral of sinx(cosx)**2 is like a**u du because you have du which is sinx * dx.
 

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