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Beliebt Rechnen >

integral of (xarccos(x))/(sqrt(1-x^2))

Lösung

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

Lösung

- arccos (x) 1 - x^{2} - x + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - cos (u) u d u

Entferne die Konstante:

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

= - \int cos (u) u d u

Wende die partielle Integration an

= - (u sin (u) - \int sin (u) d u)

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

= - (u sin (u) - (- cos (u)))

Setze in

u = arccos (x)

ein

= - (arccos (x) sin (arccos (x)) - (- cos (arccos (x))))

Vereinfache

- (arccos (x) sin (arccos (x)) - (- cos (arccos (x)))) : - arccos (x) 1 - x^{2} - x

= - arccos (x) 1 - x^{2} - x

Füge eine Konstante zur Lösung hinzu

= - arccos (x) 1 - x^{2} - x + C

Graph

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