Improper Integrals, Comparison and Laplace Transforms

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I. Evaluate the integrals or prove they diverge

1. \[\displaystyle \int_{2}^{4} \frac{dx}{(x-3)^2}\]
2. \[\displaystyle \int_{0}^{2} \frac{dx}{\sqrt{4-x^2}}\]
3. \[\displaystyle \int_{-\infty}^{\infty} \frac{dx}{x^2+2x+5}\]
4. \[\displaystyle \int_{e^2}^{\infty} \frac{dx}{x\ln^3 x}\]
5. \[\displaystyle \int_{0}^{\infty} x\sin x\,dx\]
6. \[\displaystyle \int_{0}^{\infty} e^{-x}\sin x\,dx\]

II. Test for convergence using the comparison test

1. \[\displaystyle \int_{a}^{\infty} e^{-px}\,dx,\quad p>0\]
2. \[\displaystyle \int_{0}^{\infty} \frac{dx}{1+2x^2+3x^4}\]
3. \[\displaystyle \int_{e^2}^{\infty} \frac{dx}{x+\sin^2 x}\]
4. \[\displaystyle \int_{1}^{\infty} \frac{\arctan x}{x}\,dx\]
5. \[\displaystyle \int_{2}^{\infty} \frac{\sqrt[7]{3+2x^2}}{\sqrt[5]{x^3-1}}\,dx\]
6. \[\displaystyle \int_{1}^{\infty} \left(1- \cos\frac{2}{x}\right) dx\]

III. Derive the Laplace transform of each function

IV. Rewrite the equations in terms of Laplace transforms

1. \[y' = ay\]
2. \[T' = -k(T-T_a)\]
3. \[\displaystyle \frac{du}{dt}= r_{in}\,c_{in} - r_{out}\,\frac{u(t)}{V}\]
4. \[y'' + 4y' + 4y = \cos 2t\]

V. Solve problems using the Laplace transform

Rewrite problems 8–10 from worksheet 2 (cooling and mixing) using Laplace transforms and isolate \(\mathcal{L}\{f(t)\}\).

VI. Extra problems

• a. Prove that \(\displaystyle \int_{0}^{\infty} \frac{\sin x}{x}\,dx\) converges.
• b. Compute \(\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^2+a^2}\,dx\), where \(a>0\).
• c. Give an example of a function \(f: (2, \infty) \to (0, \infty)\) such that \(\displaystyle \int_{2}^{\infty} f^p(x)\,dx\) is finite if and only if \(p\in [2, \infty)\).