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Beliebt Rechnen >

integral of (x^2+4x+9)cos(3x)

Lösung

\int (x^{2} + 4 x + 9) cos (3 x) d x

Lösung

\frac{1}{27} (9 x^{2} sin (3 x) + 6 x cos (3 x) + 36 x sin (3 x) + 79 sin (3 x) + 12 cos (3 x)) + C

Schritte zur Lösung

\int (x^{2} + 4 x + 9) cos (3 x) d x

Wende U-Substitution an

= \int \frac{1}{3} cos (u) (\frac{u ^{2}}{9} + \frac{4 u}{3} + 9) d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int cos (u) (\frac{u ^{2}}{9} + \frac{4 u}{3} + 9) d u

Multipliziere aus

cos (u) (\frac{u ^{2}}{9} + \frac{4 u}{3} + 9) : \frac{u ^{2} cos ( u )}{9} + \frac{4 u cos ( u )}{3} + 9 cos (u)

= \frac{1}{3} \cdot \int \frac{u ^{2} cos ( u )}{9} + \frac{4 u cos ( u )}{3} + 9 cos (u) d u

Wende die Summenregel an:

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

= \frac{1}{3} (\int \frac{u ^{2} cos ( u )}{9} d u + \int \frac{4 u cos ( u )}{3} d u + \int 9 cos (u) d u)

\int \frac{u ^{2} cos ( u )}{9} d u = \frac{1}{9} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

\int \frac{4 u cos ( u )}{3} d u = \frac{4}{3} (u sin (u) + cos (u))

\int 9 cos (u) d u = 9 sin (u)

= \frac{1}{3} (\frac{1}{9} (u^{2} sin (u) - 2 (- u cos (u) + sin (u))) + \frac{4}{3} (u sin (u) + cos (u)) + 9 sin (u))

Setze in

u = 3 x

ein

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

Vereinfache

\frac{1}{3} (\frac{1}{9} ((3 x)^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + \frac{4}{3} (3 x sin (3 x) + cos (3 x)) + 9 sin (3 x)) : \frac{1}{27} (9 x^{2} sin (3 x) + 6 x cos (3 x) + 36 x sin (3 x) + 79 sin (3 x) + 12 cos (3 x))

= \frac{1}{27} (9 x^{2} sin (3 x) + 6 x cos (3 x) + 36 x sin (3 x) + 79 sin (3 x) + 12 cos (3 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{27} (9 x^{2} sin (3 x) + 6 x cos (3 x) + 36 x sin (3 x) + 79 sin (3 x) + 12 cos (3 x)) + C

Graph

Beliebte Beispiele

integral of 1/(x^2+4x+10)

\int \frac{1}{x ^{2} + 4 x + 10} d x

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

integral of (20x)/(x^2+1)

\int \frac{20 x}{x ^{2} + 1} d x

integral of 1/(25+9y^2)

\int \frac{1}{25 + 9 y ^{2}} d y

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\int x^{5} 4 - x^{3} d x