Simplifying Trigonometric Expression and Solving Integral

[holezch]

Homework Statement

$$\int sin^{2} u - cos^{2} u / \sqrt{sin^{4} u + cos^{4} }$$

Homework Equations

The Attempt at a Solution

$$\int sin^{2}(u) - cos^{2}(u) / \sqrt{sin^{4}(u) + cos^{4}(u)}$$
then
$$\sqrt{sin^{4} u + cos^{4}} <br /> = \sqrt{(sin^{2}(u) + cos^{2}(u))^{2} - sin^{2}(u)cos^{2}(u)} <br /> = \sqrt{1 - 2sin^{2}(u)cos^{2}(u)}<br /> = \sqrt{\frac{1+cos^{2}(2u)}{2}} <br /> OR \sqrt{\frac{2 - 2sin^{2}(2u)}{2}}$$that's as much as I could simplify.. any help would be appreciated, thanks

 

[Dick]

Your numerator is pretty close to being cos(2u). Can you express the denominator in terms of sin(2u)? Then you should be able to see the substitution to use.

 

[holezch]

wow, the substitution has been right in front me! I've been blind..

so the denominator is $$\sqrt{\frac{2 - sin^{2}(2u)}{2}}$$

and the numerator is -cos(2u)

so I can use t = sin(2u)

thanks!