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integral de 3sin^5(x)cos^{-3/2}(x)

Solución

\int 3 sin^{5} (x) cos^{- \frac{3}{2}} (x) d x

3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x)) + C

Pasos de solución

\int 3 sin^{5} (x) cos^{- \frac{3}{2}} (x) d x

Sacar la constante:

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

= 3 \cdot \int sin^{5} (x) cos^{- \frac{3}{2}} (x) d x

- \frac{3}{2} = 3 \cdot \frac{1}{2}

= 3 \cdot \int sin^{5} (x) cos^{3 \cdot \frac{1}{2}} (x) d x

Multiplicar

3 \cdot \frac{1}{2} : \frac{3}{2}

= 3 \cdot \int sin^{5} (x) cos^{\frac{3}{2}} (x) d x

Re-escribir usando identidades trigonométricas

= 3 \cdot \int (1 - cos^{2} (x))^{2} sin (x) cos^{\frac{3}{2}} (x) d x

Aplicar integración por sustitución

: \int - u^{\frac{3}{2}} (1 - u^{2})^{2} d u

= 3 \cdot \int - u^{\frac{3}{2}} (1 - u^{2})^{2} d u

Expandir

- u^{\frac{3}{2}} (1 - u^{2})^{2} : - u^{\frac{3}{2}} + 2 u^{\frac{7}{2}} - u^{\frac{11}{2}}

= 3 \cdot \int - u^{\frac{3}{2}} + 2 u^{\frac{7}{2}} - u^{\frac{11}{2}} d u

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

= 3 (- \int u^{\frac{3}{2}} d u + \int 2 u^{\frac{7}{2}} d u - \int u^{\frac{11}{2}} d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u^{\frac{7}{2}} d u = \frac{4}{9} u^{\frac{9}{2}}

\int u^{\frac{11}{2}} d u = \frac{2}{13} u^{\frac{13}{2}}

= 3 (- \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{9} u^{\frac{9}{2}} - \frac{2}{13} u^{\frac{13}{2}})

Sustituir en la ecuación

u = cos (x)

= 3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x))

Agregar una constante a la solución

= 3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x)) + C

\int \frac{2}{x} + 3 e^{x} d x

\int \frac{1}{x ( 2 + 3 u )} d x

y^{^{''}} - 2 y^{^{'}} + y = 0

\int x (12 x + 4) d x

x \to 2 lim (ln ∣ 2 - x ∣)