What is the integral of tcos(t)e^{-st} ?

The integral of tcos(t)e^{-st} is

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integral of tcos(t)e^{-st}

\int t cos (t) e^{- s t} d t

- e^{- s t} \frac{1}{s} t cos (t) - e^{- s t} \frac{1}{s ^{2}} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) + e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) + e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) + e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) + e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t) + C

Solution steps

\int t cos (t) e^{- s t} d t

Apply Integration By Parts

: \frac{1}{s ^{2}} cos (t) (- s e^{- s t} t - e^{- s t}) - \int - \frac{1}{s ^{2}} sin (t) (- s e^{- s t} t - e^{- s t}) d t

= \frac{1}{s ^{2}} cos (t) (- s e^{- s t} t - e^{- s t}) - \int - \frac{1}{s ^{2}} sin (t) (- s e^{- s t} t - e^{- s t}) d t

\int - \frac{1}{s ^{2}} sin (t) (- s e^{- s t} t - e^{- s t}) d t = - e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) - e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) - e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) - e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) - e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t)

= \frac{1}{s ^{2}} cos (t) (- s e^{- s t} t - e^{- s t}) - (- e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) - e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) - e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) - e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) - e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t))

Simplify

\frac{1}{s ^{2}} cos (t) (- s e^{- s t} t - e^{- s t}) - (- e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) - e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) - e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) - e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) - e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t)) : - e^{- s t} \frac{1}{s} t cos (t) - e^{- s t} \frac{1}{s ^{2}} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) + e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) + e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) + e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) + e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t)

= - e^{- s t} \frac{1}{s} t cos (t) - e^{- s t} \frac{1}{s ^{2}} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) + e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) + e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) + e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) + e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t)

Add a constant to the solution

= - e^{- s t} \frac{1}{s} t cos (t) - e^{- s t} \frac{1}{s ^{2}} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} t cos (t) + e^{- s t} \frac{1}{1 + s ^{2}} t sin (t) - e^{- s t} \frac{1}{s ( s ^{2} + 1 ) ^{2}} sin (t) + e^{- s t} \frac{2}{( 1 + s ^{2} ) ^{2}} cos (t) + e^{- s t} \frac{s}{( 1 + s ^{2} ) ^{2}} sin (t) + e^{- s t} \frac{1}{s ^{2} ( 1 + s ^{2} )} cos (t) + e^{- s t} \frac{1}{s ( s ^{2} + 1 )} sin (t) + C

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