Is Using Exponent Identities Allowed in Solving Trigonometric Integrals?

[XJellieBX]

Homework Statement

Compute \int^{\pi/2}_{0} \frac{sin^{2009}x}{sin^{2009}x + cos^{2009}x}

I used the identity cos^{2}= 1 - sin^{2}, but instead I set the exponent as 2009. And so I ended up with the answer being -1. I'm just wondering whether this is a legal solution or am I not allowed to do that. Thanks.

 

[AssyriaQ]

sin^{2}x + cos^{2}x = 1 is the so called Pythagorean trigonometric identity. It is not valid when replacing the exponent 2 by another number, i.e.,

sin^{n}x + cos^{n}x \neq 1 for n\neq 2.

 

[XJellieBX]

Thank you, I really needed that second opinion =)

 

[Dick]

Homework Statement

Compute \int^{\pi/2}_{0} \frac{sin^{2009}x}{sin^{2009}x + cos^{2009}x}

I used the identity cos^{2}= 1 - sin^{2}, but instead I set the exponent as 2009. And so I ended up with the answer being -1. I'm just wondering whether this is a legal solution or am I not allowed to do that. Thanks.

Try the change of variables x -> pi/2-x to get a new integral. Then add it to the old integral.

 

[XJellieBX]

Thank you =) I found the answer.