integral of (4x^2-5x+6)sin(x)

Lösung

\int (4 x^{2} - 5 x + 6) sin (x) d x

- 4 x^{2} cos (x) + 8 x sin (x) + 5 x cos (x) + 2 cos (x) - 5 sin (x) + C

Schritte zur Lösung

\int (4 x^{2} - 5 x + 6) sin (x) d x

Multipliziere aus

(4 x^{2} - 5 x + 6) sin (x) : 4 x^{2} sin (x) - 5 x sin (x) + 6 sin (x)

= \int 4 x^{2} sin (x) - 5 x sin (x) + 6 sin (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 4 x^{2} sin (x) d x - \int 5 x sin (x) d x + \int 6 sin (x) d x

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

\int 5 x sin (x) d x = 5 (- x cos (x) + sin (x))

\int 6 sin (x) d x = - 6 cos (x)

= 4 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) - 5 (- x cos (x) + sin (x)) - 6 cos (x)

Vereinfache

4 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) - 5 (- x cos (x) + sin (x)) - 6 cos (x) : - 4 x^{2} cos (x) + 8 x sin (x) + 5 x cos (x) + 2 cos (x) - 5 sin (x)

= - 4 x^{2} cos (x) + 8 x sin (x) + 5 x cos (x) + 2 cos (x) - 5 sin (x)

Füge eine Konstante zur Lösung hinzu

= - 4 x^{2} cos (x) + 8 x sin (x) + 5 x cos (x) + 2 cos (x) - 5 sin (x) + C

\int \frac{x - 1}{3 x ^{2} - 2 x + 5} d x

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

\int 2 x^{2} + x - 5 d x

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

\int \frac{( 1 - 3 x ) ^{2}}{x ^{3}} d x