de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 1/(x^3sqrt(9-x^2))

Lösung

\int \frac{1}{x ^{3} 9 - x ^{2}} d x

Lösung

\frac{1}{216} (- cot^{2} (\frac{arcsin ( \frac{x}{3} )}{2}) + 4 ln tan (\frac{arcsin ( \frac{x}{3} )}{2}) + tan^{2} (\frac{arcsin ( \frac{1}{3} x )}{2})) + C

Schritte zur Lösung

\int \frac{1}{x ^{3} 9 - x ^{2}} d x

Trigonometrische Substitution anwenden

= \int \frac{1}{27 sin ^{3} ( u )} d u

Entferne die Konstante:

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

= \frac{1}{27} \cdot \int \frac{1}{sin ^{3} ( u )} d u

Wende U-Substitution an

= \frac{1}{27} \cdot \int \frac{( 1 + v ^{2} ) ^{2}}{4 v ^{3}} d v

Entferne die Konstante:

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

= \frac{1}{27} \cdot \frac{1}{4} \cdot \int \frac{( 1 + v ^{2} ) ^{2}}{v ^{3}} d v

Multipliziere aus

\frac{( 1 + v ^{2} ) ^{2}}{v ^{3}} : \frac{1}{v ^{3}} + \frac{2}{v} + v

= \frac{1}{27} \cdot \frac{1}{4} \cdot \int \frac{1}{v ^{3}} + \frac{2}{v} + 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}{27} \cdot \frac{1}{4} (\int \frac{1}{v ^{3}} d v + \int \frac{2}{v} d v + \int v d v)

\int \frac{1}{v ^{3}} d v = - \frac{1}{2 v ^{2}}

\int \frac{2}{v} d v = 2 ln ∣ v ∣

\int v d v = \frac{v ^{2}}{2}

= \frac{1}{27} \cdot \frac{1}{4} (- \frac{1}{2 v ^{2}} + 2 ln ∣ v ∣ + \frac{v ^{2}}{2})

Ersetze zurück

= \frac{1}{27} \cdot \frac{1}{4} - \frac{1}{2 tan ^{2} ( \frac{a r c s i n ( \frac{1}{3} x )}{2} )} + 2 ln tan (\frac{arcsin ( \frac{1}{3} x )}{2}) + \frac{tan ^{2} ( \frac{a r c s i n ( \frac{1}{3} x )}{2} )}{2}

Vereinfache

\frac{1}{27} \cdot \frac{1}{4} - \frac{1}{2 tan ^{2} ( \frac{a r c s i n ( \frac{1}{3} x )}{2} )} + 2 ln tan (\frac{arcsin ( \frac{1}{3} x )}{2}) + \frac{tan ^{2} ( \frac{a r c s i n ( \frac{1}{3} x )}{2} )}{2} : \frac{1}{216} (- cot^{2} (\frac{arcsin ( \frac{x}{3} )}{2}) + 4 ln tan (\frac{arcsin ( \frac{x}{3} )}{2}) + tan^{2} (\frac{arcsin ( \frac{1}{3} x )}{2}))

= \frac{1}{216} (- cot^{2} (\frac{arcsin ( \frac{x}{3} )}{2}) + 4 ln tan (\frac{arcsin ( \frac{x}{3} )}{2}) + tan^{2} (\frac{arcsin ( \frac{1}{3} x )}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{216} (- cot^{2} (\frac{arcsin ( \frac{x}{3} )}{2}) + 4 ln tan (\frac{arcsin ( \frac{x}{3} )}{2}) + tan^{2} (\frac{arcsin ( \frac{1}{3} x )}{2})) + C

Graph

Beliebte Beispiele

integral of (4x^3-5)

\int (4 x^{3} - 5) d x

integral of 1/(sqrt(x)(1-7\sqrt{x))}

\int \frac{1}{x ( 1 - 7 x )} d x

integral of e^{ux}

\int e^{ux} d x

integral of (e^{3+ln(x)})/(-2x)

\int \frac{e ^{3 + l n (x)}}{- 2 x} d x

integral of (21x^6+x-2e+3csc^2(x))

\int (21 x^{6} + x - 2 e + 3 csc^{2} (x)) d x