How do you use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n=6 for #int 9 sqrt (ln x) dx# from [1,4]?
See the explanation section, below.
For all three approximations, we have
The endpoints of the subintervals are:
Do the arithmetic.
Do the arithmetic.
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To approximate the integral (\int_{1}^{4} \sqrt{\ln x} , dx) using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with (n = 6), follow these steps:

Trapezoidal Rule:
[h = \frac{ba}{n} = \frac{41}{6} = \frac{1}{2}]
[x_0 = 1, x_1 = 1.5, x_2 = 2, x_3 = 2.5, x_4 = 3, x_5 = 3.5, x_6 = 4]
[T_n = \frac{h}{2}[f(x_0) + 2\sum_{i=1}^{n1}f(x_i) + f(x_n)]]

Midpoint Rule:
[m_i = \frac{x_{i1} + x_i}{2}]
[M_n = h\sum_{i=1}^{n}f(m_i)]

Simpson's Rule:
[S_n = \frac{h}{3}[f(x_0) + 4\sum_{i=1}^{n/2}f(x_{2i1}) + 2\sum_{i=1}^{n/21}f(x_{2i}) + f(x_n)]]
Now, plug in the values and calculate the approximations.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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