Evaluation of Integrals by Laplace Transforms

Example 1: Evaluate \( \int_{0}^{\infty} t e^{-2t} \sin(3t) \, dt. \)

Solution: By the definition of the Laplace transform: \[ \int_{0}^{\infty} t e^{-2t} \sin(3t) \, dt = \left[ L\{ t \sin(3t) \} \right]_{s = 2}. \] Using the multiplication by \( t \) property: \[ L\{ t \sin(3t) \} = -\frac{d}{ds} \left( L\{ \sin(3t) \} \right) = -\frac{d}{ds} \left( \frac{3}{s^2 + 9} \right). \] Evaluating the derivative: \[ L\{ t \sin(3t) \} = \frac{6s}{(s^2 + 9)^2}. \] Substituting \( s = 2 \): \[ \int_{0}^{\infty} t e^{-2t} \sin(3t) \, dt = \frac{12}{169}. \]

Example 2: Evaluate \[ \int_{0}^{\infty} \frac{e^{-\sqrt{2t}} \sinh(t) \sin(t)}{t} \, dt. \]

Solution: Using the Laplace transform definition: \[ \int_{0}^{\infty} \frac{e^{-\sqrt{2t}} \sinh(t) \sin(t)}{t} \, dt = \left[ L\left\{ \frac{\sinh(t) \sin(t)}{t} \right\} \right]_{s = \sqrt{2}}. \] Using the division by \( t \) property: \[ L\left\{ \frac{\sinh(t) \sin(t)}{t} \right\} = \int_{s}^{\infty} L\{ \sinh(t) \sin(t) \} \, ds. \] Evaluating the integral: \[ L\left\{ \frac{\sinh(t) \sin(t)}{t} \right\} = \frac{1}{2} \tan^{-1}\left( \frac{2}{s^2} \right). \] Substituting \( s = \sqrt{2} \): \[ \int_{0}^{\infty} \frac{e^{-\sqrt{2t}} \sinh(t) \sin(t)}{t} \, dt = \frac{\pi}{8}. \]

Model Problems

Evaluate the following using Laplace transforms:

1. \[ \int_{0}^{\infty} \frac{e^{-t} \sin^2(t)}{t} \, dt \]
2. \[ \int_{0}^{\infty} t e^{-2t} \cos(t) \, dt \]
3. \[ \int_{0}^{\infty} e^{-t} \frac{\cos(at) - \cos(bt)}{t} \, dt \]
4. \[ \int_{0}^{\infty} \frac{\cos(6t) - \cos(4t)}{t} \, dt \]
5. \[ \int_{0}^{\infty} \frac{\sin(mt)}{t} \, dt \]
6. \[ \int_{0}^{\infty} \frac{e^{-2t} \sinh(t)}{t} \, dt \]
7. \[ \int_{0}^{\infty} t^3 e^{-t} \sin(t) \, dt \]
8. \[ \int_{0}^{\infty} t^2 e^{-t} \cos(t) \, dt \]
9. \[ \int_{0}^{\infty} \frac{e^{-at} - e^{-bt}}{t} \, dt \]

Answers

1. \[ \frac{1}{2} \log 5 \]
2. \[ \frac{3}{25} \]
3. \[ \frac{1}{2} \log \left( \frac{1 + b^2}{1 + a^2} \right) \]
4. \[ \log \left( \frac{2}{3} \right) \]
5. \[ \frac{\pi}{2} \]
6. \[ \frac{1}{2} \log 3 \]
7. \[ 0 \]
8. \[ \frac{1}{2} \]
9. \[ \log \left( \frac{a}{b} \right) \]