evaluate $\displaystyle \int^{\frac{\pi}{3}}_0(sinx-cosx)^2 dx$
what i did
$\displaystyle \int^{\frac{\pi}{3}}_0(sinx-cosx)^2 dx=[\frac{(sinx-cosx)^3}{3(-cosx-sinx)}]^{\frac{\pi}{3}}_0$
is this right?
Sorry, I have edited my first post without actually taking a look at whether the integral was right or wrong.
My second post says it.
In fact, it's only my edit in my first post which is contradicting. The 'formula' that I posted remains the same and applies.
So, your integral is wrong. You need to put the differential of the function as denominator, not its integral.
If you had to differentiate
$\displaystyle y = (x^2 + x + 1)^3$
This becomes:
$\displaystyle \frac{dy}{dx} = (x^2 + x + 1)^2 . (2x + 1)$
When you integrate, you divide instead of multiplying the differential of the function.
$\displaystyle \int^{\frac{\pi}{3}}_0(sinx-cosx)^2 dx = [\frac{(sinx - cosx)^3}{3(cosx + sinx)}]^{\frac{\pi}{3}} _0$
$\displaystyle = \frac{(sin(\frac{\pi}{3}) - cos(\frac{\pi}{3}))^3}{3(cos(\frac{\pi}{3}) + sin(\frac{\pi}{3}))} - \frac{(sin(0) - cos(0))^3}{3(cos(0) + sin(0))}$
$\displaystyle = \frac{(\frac{\sqrt{3}}{2} - \frac{1}{2})^3}{3(\frac12 + \frac{\sqrt{3}}{2})} - \frac{(0 - 1)^3}{3(1 + 0)}$
$\displaystyle = \frac{(\frac{\sqrt{3}-1}{2})^3}{3(\frac{1+\sqrt{3}}{2})} - \frac{-1}{3}$
$\displaystyle = \frac{(\frac{(\sqrt{3}-1)^3}{8})}{\frac{3+3\sqrt{3}}{2}} + \frac{1}{3}$
$\displaystyle = \frac{6\sqrt{3}-10}{12 + 12\sqrt{3}} + \frac{1}{3}$
If I didn't make any mistake, you can now further simplify.
EDIT: Seems like it's wrong.
Maybe taking another approach to have less of those square roots...
ok, I'll try this.
$\displaystyle \int^{\frac{\pi}{3}}_0 (sin(x) - cos(x))^2 dx = \int^{\frac{\pi}{3}}_0 sin^2(x) - 2sin(x)cos(x) + cos^2(x) dx$
$\displaystyle = \int^{\frac{\pi}{3}}_0 1 - sin(2x)dx$
$\displaystyle = [x+ \frac12 cos(2x)]^{\frac{\pi}{3}}_0$
$\displaystyle = [\frac{\pi}{3} + \frac12 cos(\frac{2\pi}{3})] - [ 0 + \frac12 cos(0)]$
$\displaystyle = \frac{\pi}{3} -\frac14 - \frac12$
$\displaystyle = \frac{\pi}{3} -\frac34$
That's better