What is integral sin 2x /((sin x )^2-4)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the following substitution, such that:

`sin^2 x - 4 = t => 2 sin x*cos x dx = dt`

Using the double angle formula, yields:

`2 sin x*cos x dx = sin 2x => sin 2x dx = dt`

Replacing the variable to integrand, yields:

`int (sin 2x)/(sin^2 x - 4)dx = int (dt)/t`

`int (dt)/t = ln|t| + c`

Replacing back `sin^2 x - 4` for `t` yields:

`int (sin 2x)/(sin^2 x - 4)dx = ln|sin^2 x - 4| + c `

Since `sin^2 x <= 1` yields `|sin^2 x - 4| = ln(4 - sin^2 x)` , hence, `int (sin 2x)/(sin^2 x - 4)dx = ln(4 - sin^2 x) + c`

Hence, evaluating the given indefinite integral, using the indicated substitution, yields `int (sin 2x)/(sin^2 x - 4)dx = ln(4 - sin^2 x) + c.`

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