integral of arcsin(sqrt(x))

Lösung

\int arcsin (x) d x

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

Schritte zur Lösung

\int arcsin (x) d x

Wende U-Substitution an

= \int arcsin (u) \cdot 2 u d u

Entferne die Konstante:

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

= 2 \cdot \int arcsin (u) u d u

Wende die partielle Integration an

= 2 (\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)))

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

Setze in

u = x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e x^{2} + 4 x - 1

\int \frac{( a + b x ^{2} )}{3 a x + b x ^{3}} d x

\int_{- 1}^{0} y + 1 d y

\int \frac{1}{486} (81 - x^{2}) d x

n = 1 \sum \infty \frac{1}{n ^{2} + 4}