expand (5e-5tsin(2pit)+2pi(1-e-5t)cos(2pit))^2

الحلّ

expand

(5 e - 5 t sin (2 π t) + 2 π (1 - e - 5 t) cos (2 π t))^{2}

- 50 e t sin (2 π t) - 100 π e t cos (2 π t) - 20 π t cos (2 π t) sin (2 π t) + 20 π e t cos (2 π t) sin (2 π t) + 100 π t^{2} cos (2 π t) sin (2 π t) - 40 π^{2} t cos^{2} (2 π t) + 40 π^{2} e t cos^{2} (2 π t) + 100 π^{2} t^{2} cos^{2} (2 π t) + 25 t^{2} sin^{2} (2 π t) + 20 π e cos (2 π t) - 20 e^{2} π cos (2 π t) + 4 π^{2} cos^{2} (2 π t) - 8 π^{2} e cos^{2} (2 π t) + 4 π^{2} e^{2} cos^{2} (2 π t) + 25 e^{2}

خطوات الحلّ

(5 e - 5 t sin (2 π t) + 2 π (1 - e - 5 t) cos (2 π t))^{2}

5 e - 5 t sin (2 π t) + 2 π (1 - e - 5 t) cos (2 π t)

وسٌع:

5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t)

= (5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t))^{2}

(5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t))^{2} = (5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t)) (5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t))

= (- 5 t sin (2 π t) - 10 π t cos (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) + 5 e) (- 5 t sin (2 π t) - 10 π t cos (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) + 5 e)

(5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t)) (5 e - 5 t sin (2 π t) + 2 π cos (2 π t) - 2 π e cos (2 π t) - 10 π t cos (2 π t))

وسٌع:

- 50 e t sin (2 π t) - 100 π e t cos (2 π t) - 20 π t cos (2 π t) sin (2 π t) + 20 π e t cos (2 π t) sin (2 π t) + 100 π t^{2} cos (2 π t) sin (2 π t) - 40 π^{2} t cos^{2} (2 π t) + 40 π^{2} e t cos^{2} (2 π t) + 100 π^{2} t^{2} cos^{2} (2 π t) + 25 t^{2} sin^{2} (2 π t) + 20 π e cos (2 π t) - 20 e^{2} π cos (2 π t) + 4 π^{2} cos^{2} (2 π t) - 8 π^{2} e cos^{2} (2 π t) + 4 π^{2} e^{2} cos^{2} (2 π t) + 25 e^{2}

= - 50 e t sin (2 π t) - 100 π e t cos (2 π t) - 20 π t cos (2 π t) sin (2 π t) + 20 π e t cos (2 π t) sin (2 π t) + 100 π t^{2} cos (2 π t) sin (2 π t) - 40 π^{2} t cos^{2} (2 π t) + 40 π^{2} e t cos^{2} (2 π t) + 100 π^{2} t^{2} cos^{2} (2 π t) + 25 t^{2} sin^{2} (2 π t) + 20 π e cos (2 π t) - 20 e^{2} π cos (2 π t) + 4 π^{2} cos^{2} (2 π t) - 8 π^{2} e cos^{2} (2 π t) + 4 π^{2} e^{2} cos^{2} (2 π t) + 25 e^{2}

e x p an d e^{4 (1 + 2 t)}

e x p an d (x^{2} + 1 - x) (x^{2} - 1 + x)

e x p an d - \frac{1}{2} (1 + e^{2 x} - ln 1 + e^{2 x})

e x p an d \frac{sin ( 4 θ )}{2 cos ^{2} ( θ ) ( 4 cos ^{2} ( θ ) - 3 )}

e x p an d x + sin (y) - cos (y) d x + x (sin (y) + cos (y) d y = 0)