Calc1.60

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\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\,


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