integral of xarcsin(6x)

Lösung

\int x arcsin (6 x) d x

\frac{1}{36} (18 x^{2} arcsin (6 x) - \frac{1}{4} (arcsin (6 x) - \frac{1}{2} sin (2 arcsin (6 x)))) + C

Schritte zur Lösung

\int x arcsin (6 x) d x

Wende U-Substitution an

= \int \frac{u arcsin ( u )}{36} d u

Entferne die Konstante:

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

= \frac{1}{36} \cdot \int u arcsin (u) d u

Wende die partielle Integration an

= \frac{1}{36} (\frac{1}{2} u^{2} arcsin (u) - \int \frac{u ^{2}}{2 1 - u ^{2}} d u)

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

= \frac{1}{36} (\frac{1}{2} u^{2} arcsin (u) - \frac{1}{4} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))))

Setze in

u = 6 x

ein

= \frac{1}{36} (\frac{1}{2} (6 x)^{2} arcsin (6 x) - \frac{1}{4} (arcsin (6 x) - \frac{1}{2} sin (2 arcsin (6 x))))

\frac{1}{2} (6 x)^{2} arcsin (6 x) = 18 x^{2} arcsin (6 x)

= \frac{1}{36} (18 x^{2} arcsin (6 x) - \frac{1}{4} (arcsin (6 x) - \frac{1}{2} sin (2 arcsin (6 x))))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{36} (18 x^{2} arcsin (6 x) - \frac{1}{4} (arcsin (6 x) - \frac{1}{2} sin (2 arcsin (6 x)))) + C

\frac{d}{d x} (e^{\frac{3 x}{2}})

x \to π lim (ln (sin (x)))

\int \frac{sec ( z ) tan ( z )}{sec ( z )} d z

d er i v a t i v e (1 - 25 x^{2})^{2} arcsin (5 x)

t an g e n t \frac{64}{x ^{2} + 16}, at x = 3