How Can You Integrate sin^4(2x) Without Using the Reduction Formula?

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Discussion Overview

The discussion revolves around the integration of the function sin^4(2x) without employing the reduction formula. Participants explore various methods and techniques for solving this integral, including integration by parts and the use of double-angle formulas.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in integrating sin^4(2x) and suggests that integration by parts may be necessary.
  • Another participant proposes using double-angle formulae multiple times as a potential approach.
  • A different participant suggests that the double angle formula for cosine could be beneficial in this context.
  • One participant offers a detailed method involving substitutions and transformations, indicating a possible pathway to the solution while noting the need to track substitutions carefully.
  • A participant points out that this question has been asked previously in another thread, implying that similar discussions exist.

Areas of Agreement / Disagreement

There is no consensus on a single method for integrating sin^4(2x), as participants suggest different approaches and techniques without resolving which is the most effective.

Contextual Notes

Participants do not clarify the specific assumptions or definitions they are using, and there may be unresolved steps in the proposed methods.

ookt2c
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integrate sin^4(2x) without using the reduction formula.im stuck.
im pretty sure you have to use integration by parts.
 
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Try using double-angle formulae a couple of times.
 
The double angle formula for cos will help I believe.
 
don't know what the reduction formula is, maybe it is the trick I'm about to give you

[tex]\int sin(2x)^4 dx = \frac{1}{2} \int sin(u)^4 du = \frac{1}{2} \int (sin(u)^2)^{3/2} sin(u) du = \frac{1}{2} \int (1-cos(u)^2)^{3/2} sin(u)du = \frac{1}{2} \int (1-t^2)^{3/2} dt = \int sqrt((1-t^2)^3)[/tex]

maybe you can do this?, of cause you have to keep track of all the substitutions to get how sin and y are related but that should be possible.
 
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