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## Homework Statement

Evaluate the following integral using a change of variables:

[tex]\int\frac{dx}{\sqrt{1-\sin^4{x}}}[/tex]

## Homework Equations

If [tex]f(x)=g(u(x))u'(x)[/tex]

and [tex]\int g(x)dx = G(x) +C [/tex]

then [tex]\int f(x)dx = G(u(x))+C [/tex]

## The Attempt at a Solution

It seems helpful to first simplify a little to obtain [tex]\int\frac{dx}{\sqrt{1-\sin^4(x)}} = \int\frac{dx}{cos(x)\sqrt{1+\sin^2(x)}}[/tex]

From this, further simplification produces, [tex]\sqrt{\frac{2}{3}}\int\frac{dx}{cos(x)\sqrt{1-\frac{1}{3}\cos(2x)}}[/tex] from which I cannot determine a useful change of variable.

On another attempt, using some substitutions (leaving my work out), I obtained [tex]\frac{1}{2}\int\frac{du}{(2-u)\sqrt{u-1}\sqrt{u}}[/tex]

Hopefully I have not made any errors in my calculations. I cannot find a useful substitution from any of these steps. Is there any trick or further simplification that can be made in order to make this easier to evaluate? Thanks for the help!