English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of xsqrt(1-x^4)

Lösung

\int x 1 - x^{4} d x

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

Schritte zur Lösung

\int x 1 - x^{4} d x

Vereinfache

x^{4} : (x^{2})^{2}

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

Wende U-Substitution an

= \int \frac{1 - u ^{2}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int 1 - u^{2} d u

Trigonometrische Substitution anwenden

= \frac{1}{2} \cdot \int cos^{2} (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Wende die Summenregel an:

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

= \frac{1}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{1}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Ersetze zurück

= \frac{1}{2} \cdot \frac{1}{2} (arcsin (x^{2}) + \frac{1}{2} sin (2 arcsin (x^{2})))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t f (x) = 2 x, at x = 8

\int_{0}^{1} 2 (x^{2} + 1) e^{- x} d x

\int 4 cos (8 x) d x

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

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - 2 y = e^{2 x}