Evaluate Integral using Laplace transform

Question

I want to find the integral of the following function using Laplace Transform $$ \int_0^{\infty}\frac{\cos(6t)-\cos(4t)}{t}\,dt $$

Answer 1

Hint : write it as $$\int _0^{\infty} \frac {cos 6t- cos 4t}{t} e ^{-st} dt $$

Where s= 0

Now it become Laplace transformation

$$L(\frac {cos 6t- cos 4t}{t}) $$

We have property $$L ( \frac{ f(t) }{t}) = \int_s^{\infty}F(s) ds $$ Here $F(s) = L(f(t)) $

Using above property

$L(\frac {cos 6t- cos 4t}{t}) = \int_s^{\infty} \left(\frac{s}{s^2+6^2}-\frac{s}{s^2+4^2} \right) ds$

Integrate you will get $$=\frac{1}{2}\ln(\frac{s^2+4^2}{s^2+6^2}) $$

Now put s=0

= $$=\frac{1}{2}\ln(\frac{4^2}{6^2}) $$

$$= \ln ( 2/3) $$