What is the integral from 0 to N of e^{-st}cos(3t) ?

The integral from 0 to N of e^{-st}cos(3t) is (3e^{-Ns}sin(3N)-e^{-Ns}scos(3N)+s)/(9+s^2)

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integral from 0 to N of e^{-st}cos(3t)

\int_{0}^{N} e^{- s t} cos (3 t) d t

\frac{3 e ^{- N s} sin ( 3 N ) - e ^{- N s} s cos ( 3 N ) + s}{9 + s ^{2}}

Solution steps

\int_{0}^{N} e^{- s t} cos (3 t) d t

Compute the indefinite integral:

\int e^{- s t} cos (3 t) d t = \frac{3 e ^{- s t} sin ( 3 t )}{9 + s ^{2}} - \frac{s e ^{- s t} cos ( 3 t )}{9 + s ^{2}} + C

Compute the boundaries:

\int_{0}^{N} e^{- s t} cos (3 t) d t = \frac{3 e ^{- s N} sin ( 3 N )}{9 + s ^{2}} - \frac{s e ^{- s N} cos ( 3 N )}{9 + s ^{2}} - (- \frac{s}{s ^{2} + 9})

= \frac{3 e ^{- s N} sin ( 3 N )}{9 + s ^{2}} - \frac{s e ^{- s N} cos ( 3 N )}{9 + s ^{2}} - (- \frac{s}{s ^{2} + 9})

Simplify

= \frac{3 e ^{- N s} sin ( 3 N ) - e ^{- N s} s cos ( 3 N ) + s}{9 + s ^{2}}

y = 4 x - 3 x^{2}, (2, - 4)

(x + 2)^{2}

\frac{d y}{d x} - \frac{1}{x} y = - 9

\int \frac{5 x ^{3} + 2 x}{x - 1} d x

\frac{d}{d x} (2 + 4 cos (x^{3}) + 3 sin (x^{3}))

What is the integral from 0 to N of e^{-st}cos(3t) ?
The integral from 0 to N of e^{-st}cos(3t) is (3e^{-Ns}sin(3N)-e^{-Ns}scos(3N)+s)/(9+s^2)