integral de tan^{1/2}(x)

Solução

\int tan^{\frac{1}{2}} (x) d x

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

Passos da solução

\int tan (x) d x

Aplicar integração por substituição trigonométrica

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

Aplicar integração por substituição

= \int \frac{2 v}{4 + v ^{2}} d v

Remover a constante:

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

= 2 \cdot \int \frac{v}{4 + v ^{2}} d v

Aplicar integração por substituição

= 2 \cdot \int \frac{2 w ^{2}}{4 + w ^{4}} d w

Remover a constante:

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

= 2 \cdot 2 \cdot \int \frac{w ^{2}}{4 + w ^{4}} d w

Tirar a fração parcial de

\frac{w ^{2}}{4 + w ^{4}} : - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )}

= 2 \cdot 2 \cdot \int - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w

Aplicar a regra da soma:

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

= 2 \cdot 2 (- \int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w + \int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w)

\int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1))

\int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1))

= 2 \cdot 2 (- \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1)) + \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1)))

Substituir na equação

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

Simplificar

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

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

Adicionar uma constante à solução

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