integral de 1 a x de sqrt(cos(t)-1)

Solución

\int_{1}^{x} cos (t) - 1 d t

i (2 1 + cos (1) - 2 1 + cos (x))

Pasos de solución

\int_{1}^{x} cos (t) - 1 d t

Aplicar integración por sustitución

: \int_{c o s (1)}^{c o s (x)} - i \frac{1}{u + 1} d u

= \int_{c o s (1)}^{c o s (x)} - i \frac{1}{u + 1} d u

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - i \cdot \int_{c o s (1)}^{c o s (x)} \frac{1}{u + 1} d u

Aplicar integración por sustitución

: \int_{c o s (1) + 1}^{c o s (x) + 1} \frac{1}{v} d v

= - i \cdot \int_{c o s (1) + 1}^{c o s (x) + 1} \frac{1}{v} d v

Simplificar

\frac{1}{v} : \frac{1}{v}

= - i \cdot \int_{c o s (1) + 1}^{c o s (x) + 1} \frac{1}{v} d v

Simplificar

= - i \cdot \int_{c o s (1) + 1}^{c o s (x) + 1} \frac{1}{v} d v

Aplicar la regla de la potencia

: [2 v^{\frac{1}{2}}]_{c o s (1) + 1}^{c o s (x) + 1}

= - i [2 v^{\frac{1}{2}}]_{c o s (1) + 1}^{c o s (x) + 1}

Aplicar las leyes de los exponentes:

= - i [2 v]_{c o s (1) + 1}^{c o s (x) + 1}

Calcular los limites:

2 cos (x) + 1 - 2 cos (1) + 1

= - i (2 cos (x) + 1 - 2 cos (1) + 1)

Simplificar

= i (2 1 + cos (1) - 2 1 + cos (x))

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