integral of xsin^2(ax)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u sin ^{2} ( u )}{a ^{2}} d u

Entferne die Konstante:

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

= \frac{1}{a ^{2}} \cdot \int u sin^{2} (u) d u

Wende die partielle Integration an

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

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

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

Setze in

u = a x

ein

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

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

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

Füge eine Konstante zur Lösung hinzu

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

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

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