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. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} d x, u=\sqrt{1-x} $$

Short answer

The answer will depend on the exact approximation from the Trapezoidal Rule computation for \(n=5\). This is a numerical method and it doesn't provide a symbolic result.

Step-by-step solution

Variable Substitution

We will start by substituting the \(u\) variable into the integral. Given that \(u = \sqrt{1-x}\), we can find the substitutions for \(x\) and \(d x\). Hence, \(x = 1 - u^2\) and \(d x = -2u du\). Now we substitute these expressions into the integral and also rewrite the limits of the integral from \(x=0\) to \(1\) as \(u=1\) to \(0\): \[ \int_{1}^{0} \frac{\cos (1-u^2)}{u} * -2u \, du \]

Simplification

Before implementing the Trapezoidal Rule, simplify the integral. As you can see, the \(u\) in the denominator and the multiplied '-2u' cancel out, allowing the integral to be rewritten as: \[ -2\int_{1}^{0} \cos(1-u^2) \, du \]

Numerical Approximation with the Trapezoidal Rule

Now, use the Trapezoidal Rule with \(n=5\) to approximate the integral. It is necessary to calculate the sub-intervals and plug in the values into the formula: \[ T_n = \frac{-2}{2*5} * \left[f(1) + 2* \left(\sum_{i=1}^{4} f(x_i) \right) + f(0) \right] \] Here \(f(x)\) represents the integrand without the integration (dx) part, and the values \(x_i\) represent the values in the sub-interval. The goal now is to compute the integral using these terms.