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Chapter 6: Problem 93

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

Short Answer

Expert verified
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 by step solution

01

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 \(\int_{0}^{1} \frac{\sin (u^2)}{u} \cdot 2 \, du = 2\int_{0}^{1} u\sin (u^2) \, du\).
02

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 \(f(x)=u\sin (u^2)\).
03

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.

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Most popular questions from this chapter

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \tan ^{3}(1-x) d x $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$

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

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Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)
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