integral of arcsin(ax)

Lösung

\int arcsin (a x) d x

x arcsin (a x) + \frac{1 - a ^{2} x ^{2}}{a} + C

Schritte zur Lösung

\int arcsin (a x) d x

Wende U-Substitution an

= \int \frac{arcsin ( u )}{a} d u

Entferne die Konstante:

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

= \frac{1}{a} \cdot \int arcsin (u) d u

Wende die partielle Integration an

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

\int \frac{u}{1 - u ^{2}} d u = - 1 - u^{2}

= \frac{1}{a} (u arcsin (u) - (- 1 - u^{2}))

Setze in

u = a x

ein

= \frac{1}{a} (a x arcsin (a x) - (- 1 - (a x)^{2}))

Vereinfache

\frac{1}{a} (a x arcsin (a x) - (- 1 - (a x)^{2})) : x arcsin (a x) + \frac{1 - a ^{2} x ^{2}}{a}

= x arcsin (a x) + \frac{1 - a ^{2} x ^{2}}{a}

Füge eine Konstante zur Lösung hinzu

= x arcsin (a x) + \frac{1 - a ^{2} x ^{2}}{a} + C

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