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Beliebt Rechnen >

integral of (x^3-2sin(3x))^2

Lösung

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

Lösung

\frac{x ^{7}}{7} - 4 (- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))) + 2 (x - \frac{1}{6} sin (6 x)) + C

Schritte zur Lösung

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

Multipliziere aus

(x^{3} - 2 sin (3 x))^{2} : x^{6} - 4 x^{3} sin (3 x) + 4 sin^{2} (3 x)

= \int x^{6} - 4 x^{3} sin (3 x) + 4 sin^{2} (3 x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{6} d x - \int 4 x^{3} sin (3 x) d x + \int 4 sin^{2} (3 x) d x

\int x^{6} d x = \frac{x ^{7}}{7}

\int 4 x^{3} sin (3 x) d x = 4 (- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))))

\int 4 sin^{2} (3 x) d x = 2 (x - \frac{1}{6} sin (6 x))

= \frac{x ^{7}}{7} - 4 (- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))) + 2 (x - \frac{1}{6} sin (6 x))

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{7}}{7} - 4 (- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))) + 2 (x - \frac{1}{6} sin (6 x)) + C

Graph

Beliebte Beispiele

limit as x approaches infinity of (e^{2x}-e^x)/(ln(x+1))

x \to \infty lim (\frac{e ^{2 x} - e ^{x}}{ln ( x + 1 )})

tangent f(x)=x^3,(2,8)

t an g e n t f (x) = x^{3}, (2, 8)

derivative of 5/(2x+1)

\frac{d}{d x} (\frac{5}{2 x + 1})

derivative f(x)=3sin(x)+2cos(x)

d er i v a t i v e f (x) = 3 sin (x) + 2 cos (x)

derivative of sinh^3(2x)

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