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

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

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

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

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

\int_{- 2}^{2} 4^{x} d x

\frac{( d y )}{( d x )} = 8 x^{(3)} + 6 x^{(2)}

15 x ln (4 x)

\int \frac{8}{5 ^{3 x}} d x