integral of sinx divided by sin4x

Question

Find the solution of sin(x) divided by sin(4x)

Answer 1

sin(4x)=2sin(2x)cos(2x)=4sin(x)cos(x)(2cos²(x)-1).

sin(x)/sin(4x)=1/(8cos³(x)-4cos(x)).

The indefinite integral of sin(x)/sin(4x) can be described as the area beneath the curve; but the curve is not continuous because for x=nπ/4 where n is an integer, the expression is undefined (the denominator=0). This means that the integral will be undefined (infinite) for some values of x.

Therefore we need to evaluate ∫dx/(8cos³(x)-4cos(x)).

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