Question

inter and find the laplacs transform of 1-cost/t^2
​

Answer 1

Answer:

Let

F(s)=∫∞0f(t)e−stdt=∫∞01−costt2e−stdt.

The function f(t) satisfies the bound f(t)=O(1∧t−2), thus it is absolutely integrable and we can apply Leibniz's integral to obtain

F′′(s)=∫∞0(1−cost)e−stdt=1s−ss2+1.

Integrating and using the condition F′(∞)=0, we have

F′(s)=logs−logs2+1−−−−−√.

Thus we have

F(s)=∫{logs−logs2+1−−−−−√}ds.

The first term is easily integrated to yield slogs−s. For the second term, note that

−∫logs2+1−−−−−√ds=−slogs2+1−−−−−√+∫s2s2+1ds=−slogs2+1−−−−−√+s−arctans+C.

Combining, we obtain

F(s)=slogs−slogs2+1−−−−−√−arctans+C.

But since F(∞)=0, we must have C=π2 and therefore

F(s)=slogs−slogs2+1−−−−−√−arctans+π2=slog(ss2+1−−−−−√)+arctan(1s