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Also known as Simpson’s \(\textstyle\frac{1}{3}\) Rule is a numerical integration technique that improves upon the Trapezoidal Rule by utilizing the geometry of parabolic arcs. The number of partitions \(n\) must be even.
\(\displaystyle S_n=\frac{b-a}{3n}\big( f(x_0)+4f(x_1)+2f(x_2)+\cdots+4f(x_{n-1})+2f(x_n) \big)\)
\(\displaystyle \Delta x=\frac{b-a}{n}\quad \displaystyle x_k=a+k\Delta x\)
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Approximate the given interval using Simpson Rule using the indicated partitions \(n\).
By setting \(u=2x+1\) gives \(du=2dx\) and substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=2x+3\), \(x = \frac{u-3}{2}\), and \(du=2dx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=2x+1\), \(x = \frac{u-1}{2}\), and \(du=2dx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=x^2\), and \(du=2xdx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=2x^3+1\), and \(du=6x^2dx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=\texttt{e}^{2x}\), and \(du=2\texttt{e}^{2x}dx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=\texttt{ln}(x)\), and \(du=\frac{dx}{x}\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=\texttt{cos}(x)+1\), and \(du=-\texttt{sin}(x)dx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=\texttt{e}^x\), and \(du=\texttt{e}^xdx\). Substituting these values into the integral leads to the following:
\(\normalsizeTransform the current integrand using algebraic manipulations:
\(\normalsize
Let \(u=\texttt{e}^x\), and \(du=\texttt{e}^xdx\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=\sqrt{x}\), \(du=\frac{1}{2\sqrt{x}}dx\), and \(x+1=u^2+1\). Substituting these values into the integral leads to the following:
\(\normalsizeLet \(u=x^2\), and \(du=2xdx\). Substituting these values into the integral leads to the following:
\(\normalsizeTransform the integrand using algebraic manipulations:
\(\normalsizeCheck with your tutor
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