integral de sqrt(1+csc(x))

Solución

\int 1 + csc (x) d x

2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)) + C

Pasos de solución

\int 1 + csc (x) d x

Re-escribir usando identidades trigonométricas

= \int 1 + \frac{1}{sin ( x )} d x

Aplicar integración por sustitución

: \int \frac{2 ( u + 1 )}{u ( 1 + u ^{2} )} d u

= \int \frac{2 ( u + 1 )}{u ( 1 + u ^{2} )} d u

Sacar la constante:

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

= 2 \cdot \int \frac{u + 1}{u ( 1 + u ^{2} )} d u

Expandir

\frac{u + 1}{u ( 1 + u ^{2} )} : \frac{u}{u ( 1 + u ^{2} )} + \frac{1}{u ( 1 + u ^{2} )}

Aplicar la regla de la suma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{u}{u ( 1 + u ^{2} )} d u + \int \frac{1}{u ( 1 + u ^{2} )} d u)

\int \frac{u}{u ( 1 + u ^{2} )} d u = 22 (- \frac{1}{8} (ln 2 u + 22 u + 2 - 2 arctan (2 u + 1)) + \frac{1}{8} (ln 2 u - 22 u + 2 + 2 arctan (2 u - 1)))

\int \frac{1}{u ( 1 + u ^{2} )} d u = 2 (\frac{1}{4 2} (ln 2 u + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u - 22 u + 2 + 2 arctan (2 u - 1)))

= 2 (22 (- \frac{1}{8} (ln 2 u + 22 u + 2 - 2 arctan (2 u + 1)) + \frac{1}{8} (ln 2 u - 22 u + 2 + 2 arctan (2 u - 1))) + 2 (\frac{1}{4 2} (ln 2 u + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u - 22 u + 2 + 2 arctan (2 u - 1))))

Sustituir en la ecuación

u = tan (\frac{x}{2})

= 2 (22 (- \frac{1}{8} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 - 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{8} (ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) + 2 (\frac{1}{4 2} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{4 2} (- ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))))

Expandir

22 (- \frac{1}{8} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 - 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{8} (ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) + 2 (\frac{1}{4 2} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{4 2} (- ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) : 2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)

= 2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1))

Agregar una constante a la solución

= 2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)) + C

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