How do i integrate SinxCosX by parts?
I am stuck
Plato is right that using Integration by Parts here is overkill, but if you MUST use it...
$\displaystyle \displaystyle \begin{align*} I &= \int{\sin{(x)}\cos{(x)}\,dx} \\ I &= \sin^2{(x)} - \int{\cos{(x)}\sin{(x)}\,dx} \\ I &= \sin^2{(x)} - I \\ 2I &= \sin^2{(x)} \\ I &= \frac{1}{2}\sin^2{(x)} + C \end{align*}$
I personally prefer to avoid integration by parts where possible. This is easier integrated if you change the integrand using a double angle formula.
$\displaystyle \displaystyle \begin{align*} \int{\sin{(x)}\cos{(x)}\,dx} &= \int{\frac{1}{2}\sin{(2x)}\,dx} \\ &= -\frac{1}{4}\cos{(2x)} + C \\ &= -\frac{1}{4} \left[ 1 - 2\sin^2{(x)} \right] + C \\ &= \frac{1}{2}\sin^2{(x)} - \frac{1}{4} + C \end{align*}$
As you can see, these answers differ only by a constant, and since the constants are arbitrary anyway, they can be considered to be equivalent