integral de (6x^8)/(sec(10x^3))

Solución

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

\frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

Pasos de solución

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

Sacar la constante:

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

= 6 \cdot \int \frac{x ^{8}}{sec ( 10 x ^{3} )} d x

Aplicar integración por sustitución

: \int \frac{u ^{2}}{3000 sec ( u )} d u

= 6 \cdot \int \frac{u ^{2}}{3000 sec ( u )} d u

Sacar la constante:

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

= 6 \cdot \frac{1}{3000} \cdot \int \frac{u ^{2}}{sec ( u )} d u

Utilizar la identidad trigonométrica básica:

\frac{1}{sec ( x )} = cos (x)

= 6 \cdot \frac{1}{3000} \cdot \int u^{2} cos (u) d u

Aplicar integración por partes

: u^{2} sin (u) - \int 2 u sin (u) d u

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Sustituir en la ecuación

u = 10 x^{3}

= 6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3})))

Simplificar

6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3}))) : \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

Agregar una constante a la solución

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

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