Question

4. Laplace transform if cos(at)u(t) is?
a) s/a²+s²?
b) a/a²+s²
c) s²/a²+s²
d) a²/a²+s²​

Answer 1

Answer:

3. Laplace transform if sin(at)u(t) is?

a) s/a²+5² b) a/a²+s² c) s²/a²+5² d) a²/a²+s²

Answer 2

Given,

Laplace transform of $\[\cos \left( {at} \right)u\left( t \right)\]$

Solution,

Calculate the Laplace transform.

$\[ = L\left( {\cos \left( {at} \right)u\left( t \right)} \right)\]$

$\[ = \int\limits_0^\infty {{e^{ - st}}f\left( t \right)dt} \]$

$\[ = \int\limits_0^\infty {{e^{ - st}}\left( {\cos \left( {at} \right)u\left( t \right)} \right)dt} \]$

$\[ = \left[ {\frac{{{e^{ - st}}}}{{{s^2} + {a^2}}}\left[ { - s\cos at + a\sin at} \right]} \right]_0^\infty \]$

$\[ = \frac{{{e^{ - \infty t}}}}{{{s^2} + {a^2}}}\left[ { - s\cos a \times \infty + a\sin a \times \infty } \right] - \frac{{{e^{ - 0 \times t}}}}{{{s^2} + {a^2}}}\left[ { - s\cos a \times 0 + a\sin a \times 0} \right]\]$

$\[ = \frac{0}{{{s^2} + {a^2}}}\left[ { - s\cos a \times \infty + a\sin a \times \infty } \right] - \frac{1}{{{s^2} + {a^2}}}\left[ { - s + a \times 0} \right]\]$

$\[ = \frac{s}{{{s^2} + {a^2}}}\]$

Hence the Laplace transform is $\[\frac{s}{{{s^2} + {a^2}}}\]$

The correct option is (a), i.e. $\[\frac{s}{{{s^2} + {a^2}}}\]$