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integral de (4sin^2(3x))(2cos^4(3x))

Solución

\int (4 sin^{2} (3 x)) (2 cos^{4} (3 x)) d x

\frac{1}{36} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

Pasos de solución

\int 4 sin^{2} (3 x) \cdot 2 cos^{4} (3 x) d x

Sacar la constante:

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

= 4 \cdot 2 \cdot \int sin^{2} (3 x) cos^{4} (3 x) d x

Simplificar

= 8 \cdot \int sin^{2} (3 x) cos^{4} (3 x) d x

Aplicar integración por sustitución

: \int sin^{2} (u) cos^{4} (u) \frac{1}{3} d u

= 8 \cdot \int sin^{2} (u) cos^{4} (u) \frac{1}{3} d u

Sacar la constante:

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

= 8 \cdot \frac{1}{3} \cdot \int sin^{2} (u) cos^{4} (u) d u

Re-escribir usando identidades trigonométricas

= 8 \cdot \frac{1}{3} \cdot \int (1 - cos^{2} (u)) cos^{4} (u) d u

Expandir

(1 - cos^{2} (u)) cos^{4} (u) : cos^{4} (u) - cos^{6} (u)

= 8 \cdot \frac{1}{3} \cdot \int cos^{4} (u) - cos^{6} (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

= 8 \cdot \frac{1}{3} (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

\int cos^{4} (u) d u = cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

= 8 \cdot \frac{1}{3} (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) - (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

Sustituir en la ecuación

u = 3 x

= 8 \cdot \frac{1}{3} (cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x - \frac{1}{4} sin (4 \cdot 3 x)) - (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x)))))

Simplificar

8 \cdot \frac{1}{3} (cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x - \frac{1}{4} sin (4 \cdot 3 x)) - (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x))))) : \frac{1}{36} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

= \frac{1}{36} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

Agregar una constante a la solución

= \frac{1}{36} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

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

\int \frac{1}{4 - ( x - 1 ) ^{2}} d x

\int 10000 - 2 (2 q + q^{3}) d q

\int 20 e^{4 x} d x