integral de xsqrt(25x-8)

Solución

\int x 25 x - 8 d x

\frac{1}{625} (\frac{2}{5} (25 x - 8)^{\frac{5}{2}} + \frac{16}{3} (25 x - 8)^{\frac{3}{2}}) + C

Pasos de solución

\int x 25 x - 8 d x

Aplicar integración por sustitución

= \int \frac{u ( u + 8 )}{625} d u

Sacar la constante:

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

= \frac{1}{625} \cdot \int u (u + 8) d u

Expandir

u (u + 8) : u^{\frac{3}{2}} + 8 u

= \frac{1}{625} \cdot \int u^{\frac{3}{2}} + 8 u 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

= \frac{1}{625} (\int u^{\frac{3}{2}} d u + \int 8 u d u)

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

\int 8 u d u = \frac{16}{3} u^{\frac{3}{2}}

= \frac{1}{625} (\frac{2}{5} u^{\frac{5}{2}} + \frac{16}{3} u^{\frac{3}{2}})

Sustituir en la ecuación

u = 25 x - 8

= \frac{1}{625} (\frac{2}{5} (25 x - 8)^{\frac{5}{2}} + \frac{16}{3} (25 x - 8)^{\frac{3}{2}})

Agregar una constante a la solución

= \frac{1}{625} (\frac{2}{5} (25 x - 8)^{\frac{5}{2}} + \frac{16}{3} (25 x - 8)^{\frac{3}{2}}) + C