I want to find the integral of the following function using Laplace Transform
$$
\int_0^{\infty}\frac{\cos(6t)-\cos(4t)}{t}\,dt
$$
Evaluate Integral using Laplace transform
integrationlaplace transform
integrationlaplace transform
I want to find the integral of the following function using Laplace Transform
$$
\int_0^{\infty}\frac{\cos(6t)-\cos(4t)}{t}\,dt
$$
Best Answer
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
= $$=\frac{1}{2}\ln(\frac{4^2}{6^2}) $$
$$= \ln ( 2/3) $$