What is (i(e^{2+3i}-e^{-2-3i}))/2 ?

The solution to (i(e^{2+3i}-e^{-2-3i}))/2 is (-sin(3)-e^4sin(3))/(2e^2)+i(-cos(3)+e^4cos(3))/(2e^2)

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simplify (i(e^{2+3i}-e^{-2-3i}))/2

simplify

\frac{i ( e ^{2 + 3 i} - e ^{- 2 - 3 i} )}{2}

\frac{- sin ( 3 ) - e ^{4} sin ( 3 )}{2 e ^{2}} + i \frac{- cos ( 3 ) + e ^{4} cos ( 3 )}{2 e ^{2}}

Solution steps

\frac{i ( e ^{2 + 3 i} - e ^{- 2 - 3 i} )}{2}

i (e^{2 + 3 i} - e^{- 2 - 3 i}) = i (e^{2} (cos (3) + i sin (3)) - e^{- 2} (cos (3) - i sin (3)))

= \frac{i ( - e ^{- 2} ( - i sin ( 3 ) + cos ( 3 ) ) + e ^{2} ( cos ( 3 ) + i sin ( 3 ) ) )}{2}

i (e^{2} (cos (3) + sin (3) i) - e^{- 2} (cos (3) - sin (3) i)) : - \frac{e ^{4} sin ( 3 ) + sin ( 3 )}{e ^{2}} + i \frac{e ^{4} cos ( 3 ) - cos ( 3 )}{e ^{2}}

= \frac{- \frac{e ^{4} s i n ( 3 ) + s i n ( 3 )}{e ^{2}} + i \frac{e ^{4} c o s ( 3 ) - c o s ( 3 )}{e ^{2}}}{2}

Multiply

\frac{e ^{4} cos ( 3 ) - cos ( 3 )}{e ^{2}} i : \frac{i ( e ^{4} cos ( 3 ) - cos ( 3 ) )}{e ^{2}}

= \frac{- \frac{e ^{4} s i n ( 3 ) + s i n ( 3 )}{e ^{2}} + \frac{i ( e ^{4} c o s ( 3 ) - c o s ( 3 ) )}{e ^{2}}}{2}

Combine the fractions

- \frac{e ^{4} sin ( 3 ) + sin ( 3 )}{e ^{2}} + \frac{i ( e ^{4} cos ( 3 ) - cos ( 3 ) )}{e ^{2}} : \frac{- ( e ^{4} sin ( 3 ) + sin ( 3 ) ) + i ( e ^{4} cos ( 3 ) - cos ( 3 ) )}{e ^{2}}

= \frac{\frac{- ( e ^{4} s i n ( 3 ) + s i n ( 3 ) ) + i ( e ^{4} c o s ( 3 ) - c o s ( 3 ) )}{e ^{2}}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- ( e ^{4} sin ( 3 ) + sin ( 3 ) ) + ( e ^{4} cos ( 3 ) - cos ( 3 ) ) i}{e ^{2} \cdot 2}

Expand

- (e^{4} sin (3) + sin (3)) + (e^{4} cos (3) - cos (3)) i : - e^{4} sin (3) - sin (3) + e^{4} i cos (3) - i cos (3)

= \frac{- e ^{4} sin ( 3 ) - sin ( 3 ) + e ^{4} i cos ( 3 ) - i cos ( 3 )}{2 e ^{2}}

Rewrite

\frac{- e ^{4} sin ( 3 ) - sin ( 3 ) + e ^{4} i cos ( 3 ) - i cos ( 3 )}{e ^{2} \cdot 2}

in standard complex form:

\frac{- e ^{4} sin ( 3 ) - sin ( 3 )}{2 e ^{2}} + \frac{e ^{4} cos ( 3 ) - cos ( 3 )}{2 e ^{2}} i

= \frac{- e ^{4} sin ( 3 ) - sin ( 3 )}{2 e ^{2}} + \frac{e ^{4} cos ( 3 ) - cos ( 3 )}{2 e ^{2}} i

x - y d x + (x + y) d y

(- 10, 0, 0) (- 6, - 9, - in f ini t y)

\frac{25 ( - 15 + 2 \cdot 2 ^{\frac{2}{3}} )}{4 \cdot 2 ^{\frac{1}{2}}}

\frac{s + 1}{s ( s + 3 )}

\frac{d ^{2}}{d x ^{2}} ((y (A, x, B, C)) (A e^{3 x} + B e^{- x} + C e^{x}))

What is (i(e^{2+3i}-e^{-2-3i}))/2 ?
The solution to (i(e^{2+3i}-e^{-2-3i}))/2 is (-sin(3)-e^4sin(3))/(2e^2)+i(-cos(3)+e^4cos(3))/(2e^2)