laplacetransform (2te^{-3t})/5-3tsin(t)

解答

拉普拉斯变换

\frac{2 t e ^{- 3 t}}{5} - 3 t sin (t)

\frac{2}{5 ( s + 3 ) ^{2}} - \frac{6 s}{( s ^{2} + 1 ) ^{2}}

求解步骤

L {\frac{2 e ^{- 3 t} t}{5} - 3 t sin (t)}

利用拉普拉斯变换的线性特性:

对于函数

f (t), g (t)

和常数

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {\frac{2 t e ^{- 3 t}}{5}} - 3 L {t sin (t)}

L {\frac{2 t e ^{- 3 t}}{5}} : \frac{2}{5 ( s + 3 ) ^{2}}

L {t sin (t)} : \frac{2 s}{( s ^{2} + 1 ) ^{2}}

= \frac{2}{5 ( s + 3 ) ^{2}} - 3 \cdot \frac{2 s}{( s ^{2} + 1 ) ^{2}}

整理

\frac{2}{5 ( s + 3 ) ^{2}} - 3 \frac{2 s}{( s ^{2} + 1 ) ^{2}} : \frac{2}{5 ( s + 3 ) ^{2}} - \frac{6 s}{( s ^{2} + 1 ) ^{2}}

= \frac{2}{5 ( s + 3 ) ^{2}} - \frac{6 s}{( s ^{2} + 1 ) ^{2}}

\int x (x^{2} - 4 x) d x

\int e^{5 x} x^{2} d x

\int \frac{4 x ^{2}}{x ^{3} + 4} d x

\int t^{2} e^{- t} d t

x \to 2 lim (\frac{1}{x ^{2} - 1})