Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=∫0∞e−stf(t)dt where we assume s is a positive real number. For example, to find the Laplace transform of f(t)=e−t, the following improper integral is evaluated using integration by parts: F(s)=∫0∞e−ste−tdt=∫0∞e−(s+1)tdt=1(s+1) Verify the following Laplace transforms, where u is a real number. f(t)=1→F(s)=1s

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stuth1 · Accepted Answer

Step 1  f(t)=1→F(s)=1s  Step 2  Given a function f(t), the Laplace transform is a new function F(s) is defined by   F(s)=∫0∞e−stf(t)dt  where we assume s is a positive real number.  For f(t)=1, the Laplace transform is given by  F(s)=∫0∞e−st⋅1dt,  Let u=−st⇒du=−sdt⇒dt=−dus  For t=0,u=−s⋅0=0.  For t=∞,u=−s⋅∞=−∞  Therefore,  F(s)=−∫0−∞(eu)sdu  =∫−∞0(eu)sdu[∵∫abf(x)dx=−∫baf(x)dx]  =1s∫−∞0(eu)du  =1s[eu]−∞0  =1s[e0−e−∞]  =1s[1−0]  =1s  Hence, verified.

Laplace transforms A powerful tool in solving problems inengineering and physics is the Laplace transform.

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