2sin^2(1/2 x)+C=2cos(x/2)sin(x/2)+C

Solución

2 sin^{2} (\frac{1}{2} x) + C = 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

x = 4 πn, x = 2 π + 4 πn, x = \frac{π}{2} + 2 πn

Pasos de solución

2 sin^{2} (\frac{1}{2} x) + C = 2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

Restar

2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

de ambos lados

2 sin^{2} (\frac{x}{2}) - 2 cos (\frac{x}{2}) sin (\frac{x}{2}) = 0

Factorizar

2 sin^{2} (\frac{x}{2}) - 2 cos (\frac{x}{2}) sin (\frac{x}{2}) : 2 sin (\frac{x}{2}) (sin (\frac{x}{2}) - cos (\frac{x}{2}))

2 sin (\frac{x}{2}) (sin (\frac{x}{2}) - cos (\frac{x}{2})) = 0

Resolver cada parte por separado

sin (\frac{x}{2}) = 0 or sin (\frac{x}{2}) - cos (\frac{x}{2}) = 0

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) - cos (\frac{x}{2}) = 0 : x = \frac{π}{2} + 2 πn

Combinar toda las soluciones

x = 4 πn, x = 2 π + 4 πn, x = \frac{π}{2} + 2 πn

sin (x) = 7

sin^{2} (x) cos^{2} (x) = \frac{2 - 2}{16}

(sin (x) - 1) (sin (x) - 5) = 0

(4 sin^{2} (x) - 1) (cos^{2} (x) - 1) = 0

3 sin^{2} (x) - 6 sin (x) = - 3, 0 \leq x \leq 2 π