Problem 93 - Essential Calculus

Question

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x} $$

Step-by-Step Solution

Answer

Following the steps above will give us an approximation of our integral using the Trapezoidal Rule. The actual computation would be tedious and not suitable for a short answer, but executng the steps will yield the required result.

Step 1: U-Substitution

The first step is to do the u-substitution, \(u=\sqrt{x}\). This means \(du=\frac{1}{2\sqrt{x}}dx\) or \(2\,du= dx/{\sqrt{x}}\). So, this makes our integral become \(\int_{0}^{1} \frac{\sin (u^2)}{u} \cdot 2 \, du = 2\int_{0}^{1} u\sin (u^2) \, du\).

Step 2: Trapezoidal Rule

The Trapezoidal Rule of numerical integration can be used to approximate an integral, given by the formula: \[\frac{b - a}{2n} [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)],\] where \(n\) is the number of trapezoids (in this case, 5), and \(x_i\) roams over our interval between \(0\) and \(1\). In our case \(f(x)=u\sin (u^2)\).

Step 3: Apply Trapezoidal Rule

We apply the Trapezoidal Rule with \(n=5\). Our \(x_i\) values using the interval of \([0,1]\) would be \(0, 0.2, 0.4, 0.6, 0.8, 1\). To use the trapezoidal rule, we insert these values into \(f(x)\), sum them up, and apply the trapezoidal rule formula, yielding our approximation of the integral.