de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of \sqrt[4]{(1+x^{1/2})^3}

Lösung

\int 4 (1 + x^{\frac{1}{2}})^{3} d x

Lösung

\frac{8}{77} (7 x - 4) (x + 1)^{\frac{7}{4}} + C

Schritte zur Lösung

\int 4 (1 + x^{\frac{1}{2}})^{3} d x

Wende U-Substitution an

= \int \frac{2 ( 3 u - 1 )}{3 u ^{\frac{5}{12}}} d u

Entferne die Konstante:

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

= \frac{2}{3} \cdot \int \frac{3 u - 1}{u ^{\frac{5}{12}}} d u

Multipliziere aus

\frac{3 u - 1}{u ^{\frac{5}{12}}} : \frac{1}{u ^{\frac{1}{12}}} - \frac{1}{u ^{\frac{5}{12}}}

= \frac{2}{3} \cdot \int \frac{1}{u ^{\frac{1}{12}}} - \frac{1}{u ^{\frac{5}{12}}} 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

= \frac{2}{3} (\int \frac{1}{u ^{\frac{1}{12}}} d u - \int \frac{1}{u ^{\frac{5}{12}}} d u)

\int \frac{1}{u ^{\frac{1}{12}}} d u = \frac{12}{11} u^{\frac{11}{12}}

\int \frac{1}{u ^{\frac{5}{12}}} d u = \frac{12}{7} u^{\frac{7}{12}}

= \frac{2}{3} (\frac{12}{11} u^{\frac{11}{12}} - \frac{12}{7} u^{\frac{7}{12}})

Setze in

u = (1 + x^{\frac{1}{2}})^{3}

ein

= \frac{2}{3} (\frac{12}{11} ((1 + x^{\frac{1}{2}})^{3})^{\frac{11}{12}} - \frac{12}{7} ((1 + x^{\frac{1}{2}})^{3})^{\frac{7}{12}})

Vereinfache

\frac{2}{3} (\frac{12}{11} ((1 + x^{\frac{1}{2}})^{3})^{\frac{11}{12}} - \frac{12}{7} ((1 + x^{\frac{1}{2}})^{3})^{\frac{7}{12}}) : \frac{8}{77} (7 x - 4) (x + 1)^{\frac{7}{4}}

= \frac{8}{77} (7 x - 4) (x + 1)^{\frac{7}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{8}{77} (7 x - 4) (x + 1)^{\frac{7}{4}} + C

Graph

Beliebte Beispiele

integral of 1/(e^xsqrt(4+e^{2x))}

\int \frac{1}{e ^{x} 4 + e ^{2 x}} d x

integral of (3-y)cos(x)

\int (3 - y) cos (x)

integral of (x+4)/(2x^3-18x)

\int \frac{x + 4}{2 x ^{3} - 18 x} d x

integral of xsin(6x^2)

\int x sin (6 x^{2}) d x

integral of (sec(x))/(1+tan(x))

\int \frac{sec ( x )}{1 + tan ( x )} d x