integral of (x+1)sin^2(x)

Lösung

\int (x + 1) sin^{2} (x) d x

\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + \frac{1}{2} (x - \frac{1}{2} sin (2 x)) + C

Schritte zur Lösung

\int (x + 1) sin^{2} (x) d x

Multipliziere aus

(x + 1) sin^{2} (x) : x sin^{2} (x) + sin^{2} (x)

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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