# Calc1.60

$\int x^{4}\sin x\,dx\,$
Here, we have $x^{4}$, which tells us we will need to use the method of integration by parts 4 times. Luckily, there is a quicker way for integrals involving powers of $x$ multiplied by a function such as $\cos x$ or $\sin x$ or $e^{x}$. The method is called the tabular method. Make a chart with three columns. The first column will contain alternating +, -, +, ... starting with +. The second column will start out with $u=x^{4}$ and each row below will contain the derivative of the function above. Keep going until you reach 0 as a derivative. The third column contains $dv=\sin x$ and every subsequent row contains the integral of the function below.
${\begin{matrix}+\ \longrightarrow &x^{4}&\ &\sin x\\\ &\ &\searrow &\ \\-\ \longrightarrow &4x^{3}&\ &-\cos x\\\ &\ &\searrow &\ \\+\ \longrightarrow &12x^{2}&\ &-\sin x\\\ &\ &\searrow &\ \\-\ \longrightarrow &24x&\ &\cos x\\\ &\ &\searrow &\ \\+\ \longrightarrow &24&\ &\sin x\\\ &\ &\searrow &\\\ &0&\ &-\cos x\\\end{matrix}}\,$
So we have the answer already. When multiplying in the first row, as you can see by the arrows, multiply the $+$ by the $x^{4}$ in the same row and by the $-\cos x$ in the row below. Do all of these multiplications and add them together to get the answer.
$\int x^{4}\sin x\,dx=-x^{4}\cos x+4x^{3}\sin x+12x^{2}\cos x-24x\sin x-24\cos x+C\,$