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# Approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with $n=4$ Compare these results with the approximation of the integral using a graphing utility.$\displaystyle\int_{0}^{\pi/4}x \tan x \ dx$

Solution

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Answered 2 years ago
Answered 2 years ago
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Given:

$\int_0^{\pi/4} x\tan x\: dx$

We need to evaluate the integral using the trapezoidal rule and the Simpson's rule, while comparing the results with the result of a graphing calculator.

\begin{aligned} \text{Trapezoidal Rule } \\\int_a^b f(x)\: dx&\approx \frac{b-a}{2n}\left[\begin{matrix} f(x_0)+2f(x_1)+2f(x_2)\\ +..+2f(x_{n-1})+f(x_n) \end{matrix}\right] \\ \\ \text{Simpson's Rule } \\\int_a^b f(x)\: dx&\approx \frac{b-a}{3n}\left[\begin{matrix} f(x_0)+4f(x_1)+2f(x_2)\\ +..+4f(x_{n-1})+f(x_n) \end{matrix}\right] \end{aligned}

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