de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of xarcsin(x)+sqrt(1-x^2)

Lösung

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

Lösung

\frac{1}{2} arcsin (x) + \frac{3}{8} sin (2 arcsin (x)) - \frac{1}{4} arcsin (x) cos (2 arcsin (x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

cos (u) (cos (u) + u sin (u)) : cos^{2} (u) + u cos (u) sin (u)

= \int cos^{2} (u) + u cos (u) sin (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{2} (u) d u + \int u cos (u) sin (u) d u

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

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

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

Setze in

u = arcsin (x)

ein

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

Vereinfache

\frac{1}{2} (arcsin (x) + \frac{1}{2} sin (2 arcsin (x))) + \frac{1}{2} (- \frac{1}{2} arcsin (x) cos (2 arcsin (x)) + \frac{1}{4} sin (2 arcsin (x))) : \frac{1}{2} arcsin (x) + \frac{3}{8} sin (2 arcsin (x)) - \frac{1}{4} arcsin (x) cos (2 arcsin (x))

= \frac{1}{2} arcsin (x) + \frac{3}{8} sin (2 arcsin (x)) - \frac{1}{4} arcsin (x) cos (2 arcsin (x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} arcsin (x) + \frac{3}{8} sin (2 arcsin (x)) - \frac{1}{4} arcsin (x) cos (2 arcsin (x)) + C

Graph

Beliebte Beispiele

integral of x^{1/3}+5

\int x^{\frac{1}{3}} + 5 d x

integral of (6x+29)^{-2}

\int (6 x + 29)^{- 2} d x

integral of (e^{ax})/a

\int \frac{e ^{a x}}{a} d x

integral of sin^2((npi)/a x)

\int sin^{2} (\frac{nπ}{a} x) d x

integral of (x^2)/((9-9x^2)^{3/2)}

\int \frac{x ^{2}}{( 9 - 9 x ^{2} ) ^{\frac{3}{2}}} d x