integral de (sin(3θ))/(8+cos^2(3θ))

Solución

\int \frac{sin ( 3 θ )}{8 + cos ^{2} ( 3 θ )} d θ

- \frac{1}{6 2} arctan (\frac{1}{2 2} cos (3 θ)) + C

Pasos de solución

\int \frac{sin ( 3 θ )}{8 + cos ^{2} ( 3 θ )} d θ

Aplicar integración por sustitución

: \int \frac{sin ( u )}{3 ( 8 + cos ^{2} ( u ) )} d u

= \int \frac{sin ( u )}{3 ( 8 + cos ^{2} ( u ) )} d u

Sacar la constante:

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

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

Aplicar integración por sustitución

: \int - \frac{1}{8 + v ^{2}} d v

= \frac{1}{3} \cdot \int - \frac{1}{8 + v ^{2}} d v

Sacar la constante:

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

= \frac{1}{3} (- \int \frac{1}{8 + v ^{2}} d v)

Aplicar integración por sustitución

= \frac{1}{3} (- \int \frac{1}{2 2 ( w ^{2} + 1 )} d w)

Sacar la constante:

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

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

Aplicar la regla de integración:

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

= \frac{1}{3} (- \frac{1}{2 2} arctan (w))

Sustitución hacia atrás

= \frac{1}{3} (- \frac{1}{2 2} arctan (\frac{cos ( 3 θ )}{2 2}))

Simplificar

\frac{1}{3} (- \frac{1}{2 2} arctan (\frac{cos ( 3 θ )}{2 2})) : - \frac{1}{6 2} arctan (\frac{1}{2 2} cos (3 θ))

= - \frac{1}{6 2} arctan (\frac{1}{2 2} cos (3 θ))

Agregar una constante a la solución

= - \frac{1}{6 2} arctan (\frac{1}{2 2} cos (3 θ)) + C

\int (12 (3 x + 1)^{3}) d x

\int \frac{1}{5 - t} d t

\int sin (ln (w)) d w

\int \frac{x e ^{x}}{64 + e ^{x}} d x

\int (x^{2} e^{- \frac{x}{2}}) d x