integral de x^2sin(3x+2)

Solución

\int x^{2} sin (3 x + 2) d x

- \frac{1}{6} x^{2} cos (6 x) + \frac{1}{108} (6 x sin (6 x) + cos (6 x)) + C

Pasos de solución

\int x^{2} sin (3 x + 2) d x

Re-escribir usando identidades trigonométricas

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

Sacar la constante:

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

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

Re-escribir usando identidades trigonométricas

= 2 \cdot \int \frac{1}{2} x^{2} sin (6 x) d x

Sacar la constante:

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

= 2 \cdot \frac{1}{2} \cdot \int x^{2} sin (6 x) d x

Aplicar integración por partes

: - \frac{1}{6} x^{2} cos (6 x) - \int - \frac{1}{3} x cos (6 x) d x

= 2 \cdot \frac{1}{2} (- \frac{1}{6} x^{2} cos (6 x) - \int - \frac{1}{3} x cos (6 x) d x)

\int - \frac{1}{3} x cos (6 x) d x = - \frac{1}{108} (6 x sin (6 x) + cos (6 x))

= 2 \cdot \frac{1}{2} (- \frac{1}{6} x^{2} cos (6 x) - (- \frac{1}{108} (6 x sin (6 x) + cos (6 x))))

Simplificar

2 \cdot \frac{1}{2} (- \frac{1}{6} x^{2} cos (6 x) - (- \frac{1}{108} (6 x sin (6 x) + cos (6 x)))) : - \frac{1}{6} x^{2} cos (6 x) + \frac{1}{108} (6 x sin (6 x) + cos (6 x))

= - \frac{1}{6} x^{2} cos (6 x) + \frac{1}{108} (6 x sin (6 x) + cos (6 x))

Agregar una constante a la solución

= - \frac{1}{6} x^{2} cos (6 x) + \frac{1}{108} (6 x sin (6 x) + cos (6 x)) + C

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