How do I evaluate this trigonometric integral involving cosine?

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SUMMARY

The integral to evaluate is ∫(π/6 to π/4) [(sin(2x))^4 / sqrt(1 - cos(2x)) dx]. The solution approach involves multiplying by the conjugate sqrt(1 + cos(2x)) / sqrt(1 + cos(2x)) to simplify the expression. Additionally, splitting (sin(2x))^4 into (1 - (cos(2x))^2)^2 is a recommended strategy. For further insights, reviewing a related discussion on Physics Forums is advised.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin and cos functions.
  • Familiarity with integral calculus and techniques for evaluating definite integrals.
  • Knowledge of algebraic manipulation, including the use of conjugates.
  • Experience with simplifying expressions involving square roots and powers.
NEXT STEPS
  • Review the use of trigonometric identities in integral calculus.
  • Learn about the technique of multiplying by the conjugate in integrals.
  • Explore the method of splitting powers of trigonometric functions for simplification.
  • Investigate related discussions on Physics Forums for additional problem-solving strategies.
USEFUL FOR

Students studying calculus, particularly those focusing on trigonometric integrals, and educators seeking to enhance their teaching methods in integral evaluation.

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Homework Statement


evaluate the integral.


Homework Equations


integral (pi/6 to pi/4) [(sin2x)^4 / sqrt(1-cos2x) dx]


The Attempt at a Solution


1. multiplying by conjugate: sqrt(1+cos2x)/sqrt(1+cos2x) .
2. should i split the (sin2x)^4 into (1-(cos2x)^2)^2 ?
 
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