integral of x*arccos(x)

Lösung

\int x \cdot arccos (x) d x

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

Schritte zur Lösung

\int x arccos (x) d x

Wende die partielle Integration an

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

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

= \frac{1}{2} x^{2} arccos (x) - (- \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))))

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1}{x} cos (x) d x

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

\int 4 z e^{- 4 yz} d y

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

\int 2 x (3 x^{2} + 5) d x