Can You Prove the Reduction Formula for Integral of Sin(nx) Over Sin(x)?

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SUMMARY

The discussion centers on proving the reduction formula for the integral of sin(nx) over sin(x), specifically the formula In = 2sin((n-1)x)/(n-1) + In-2 for all integers n ≥ 1. Participants suggest using mathematical induction as a primary method for proof. The discussion highlights the importance of breaking down sin(nx) using the angle addition formula: sin(nx) = sin((n-1)x)·cos(x) + cos((n-1)x)·sin(x) as a foundational step in the proof process.

PREREQUISITES
  • Understanding of integral calculus, specifically integration techniques involving trigonometric functions.
  • Familiarity with mathematical induction as a proof technique.
  • Knowledge of trigonometric identities, particularly the angle addition formulas.
  • Basic skills in manipulating algebraic expressions involving sine functions.
NEXT STEPS
  • Study the principles of mathematical induction in depth.
  • Review trigonometric identities, focusing on the angle addition formulas.
  • Practice integration techniques involving trigonometric functions, particularly sin(nx).
  • Explore additional reduction formulas for integrals involving trigonometric functions.
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced calculus, particularly those studying integral calculus and trigonometric integrals.

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If I n =integral sin (nx) dx/sinx,prove that In =2sin((n-1)x)/n-1 +In-2 for all integers n>(egual) .


I don't know how to start...
 
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Since you are given the formula, the obvious approach is to use induction.
 
e179285 said:
If I n =integral sin (nx) dx/sinx,prove that In =2sin((n-1)x)/n-1 +In-2 for all integers n>(equal) .

I don't know how to start...
sin(nx) = sin((n-1)x+x)
=sin((n-1)x)∙cos(x) + cos((n-1)x)∙sin(x)​

is a good place to start.
 

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