integral of xcos^2(3x)

Lösung

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

\frac{1}{72} (18 x^{2} + 6 x sin (6 x) + cos (6 x)) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1}{2} x (1 + cos (6 x)) d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int x (1 + cos (6 x)) d x

Wende die partielle Integration an

= \frac{1}{2} (x (x + \frac{1}{6} sin (6 x)) - \int x + \frac{1}{6} sin (6 x) d x)

\int x + \frac{1}{6} sin (6 x) d x = \frac{x ^{2}}{2} - \frac{1}{36} cos (6 x)

= \frac{1}{2} (x (x + \frac{1}{6} sin (6 x)) - (\frac{x ^{2}}{2} - \frac{1}{36} cos (6 x)))

Vereinfache

\frac{1}{2} (x (x + \frac{1}{6} sin (6 x)) - (\frac{x ^{2}}{2} - \frac{1}{36} cos (6 x))) : \frac{1}{72} (18 x^{2} + 6 x sin (6 x) + cos (6 x))

= \frac{1}{72} (18 x^{2} + 6 x sin (6 x) + cos (6 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{72} (18 x^{2} + 6 x sin (6 x) + cos (6 x)) + C

\int (1 - t)^{2} d t

\frac{d}{d x} ((x^{3} - x) (2 x - x^{2}))

2 y + \frac{d y}{d x} = y^{2} + 1

h (x) = \frac{3}{x ^{2}} + 10 x^{3} - 2 sin (3 x)

\int \frac{2 + 3 x}{x - 3} d x