1. ## Incorrect integral

Hi, I was doing a question with this simple integral

$\displaystyle 3a\int^{\frac{\pi}{2}}_0cosxsinxdx$

so rather than do it by parts (which probably would turn out to be double back on itself so you get the integral again), I opted for using sin2x = 2sinxcosx

so that means the integral simplified to

$\displaystyle 3a\int^{\frac{\pi}{2}}_0\frac{1}{2}sin2x$

which integrates to

$\displaystyle 3a[-\frac{1}{4}cos2x]^{\frac{\pi}{2}}_0$

so wack in the limits and I get 3a[1-1] = 0

but that isn't right is it?

The answer scheme suggest $\displaystyle 3a\int^{\frac{\pi}{2}}_0cosxsinxdx$ = $\displaystyle 3a[\frac{1}{2}sin^{2}2x]^{\frac{\pi}{2}}_0$

but i) How come my method doesn't work
ii) How can the mark scheme do that. You can't just do that!?

Thanks

2. Originally Posted by thomas49th

$\displaystyle 3a[-\frac{1}{4}cos2x]^{\frac{\pi}{2}}_0$

so wack in the limits and I get 3a[1-1] = 0

$\displaystyle 3a[-\frac{1}{4}cos2x]^{\frac{\pi}{2}}_0$

$\displaystyle =-\frac{3a}{4}[cos2x]^{\frac{\pi}{2}}_0$

$\displaystyle =-\frac{3a}{4}[cos{\pi}-cos(0)]$

$\displaystyle =-\frac{3a}{4}[-1-1]$

$\displaystyle =\frac{3a}{2}$

3. Originally Posted by thomas49th

The answer scheme suggest $\displaystyle 3a\int^{\frac{\pi}{2}}_0cosxsinxdx$ = $\displaystyle 3a[\frac{1}{2}sin^{2}2x]^{\frac{\pi}{2}}_0$

ii) How can the mark scheme do that. You can't just do that!?

Thanks
It is incorrect

it should be
$\displaystyle 3a[\frac{1}{2}sin^{2}x]^{\frac{\pi}{2}}_0$

4. im an idiot...

how did they get there stupid $\displaystyle 3a[\frac{1}{2}sin^{2}2x]^{\frac{\pi}{2}}_0$

thanks

5. The simplest way to do $\displaystyle \int sin(x)cos(x) dx$ is neither "by parts" or using a trig identity like that. Let u= sin(x) so that du= cos(x) dx and this becomes the simple integral $\displaystyle \int u du$.

An integral of that is $\displaystyle \frac{1}{2} u^2$. That's where they get $\displaystyle 3a\left[\frac{1}{2}sin^2(x)\right]_0^{\pi/2}$.

6. oh course the one of the products is the differential of the other. cheers