This integral does not converge using the normal (Leibniz) definition of integration. Using the identity:
int(0-->inf)f(t)/(t)dt = int(0-->inf)F(p)dp, find:
integral(0-->inf)(cos(at)+cos(bt))/t dt
This very problem is discussed in Wikipedia's article on Laplace Transforms. It's clear and well done.
To evaluate the integral ∫(0 → ∞) (cos(at) + cos(bt))/t dt using the given integral identity, we can express the integrand as a function of a new variable p.
Let's consider the function f(t) = cos(at) + cos(bt) and rewrite it:
cos(at) + cos(bt) = 2cos((at + bt)/2)cos((bt - at)/2)
Now, we divide both sides by t:
(cos(at) + cos(bt))/t = 2cos((at + bt)/2)/(2t) * cos((bt - at)/2)
We can see that the right-hand side of this equation can be expressed as a function of p using the identity provided. Let's define:
F(p) = cos((at + bt)/2)/(2t)
Now, we need to determine the relationship between p and t. From the given integral identity:
∫(0 → ∞) f(t)/t dt = ∫(0 → ∞) F(p) dp
Comparing the expressions, we have:
f(t)/t = F(p)
Dividing both sides by f(t) and multiplying by t, we get:
t = f(t)/F(p)
Now, we substitute the expression for t we found into the original integral:
∫(0 → ∞) (cos(at) + cos(bt))/t dt = ∫(0 → ∞) (cos(at) + cos(bt))/(f(t)/F(p)) dt
Replacing cos(at) + cos(bt)/t with F(p), we have:
∫(0 → ∞) F(p) dt = ∫(0 → ∞) F(p) dp
Since the integral on the left converges, we can evaluate it using the integral identity. We need to find the antiderivative of F(p), which we can differentiate with respect to p to obtain F(p). We can then integrate F(p) dp from 0 to ∞ to find the value of the integral.
However, due to the complexity and lack of analytical expression for the antiderivative of F(p), this integral may not be easily evaluated. It may require numerical methods, such as numerical integration or approximation techniques, to find an approximate value.