de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral ((sqrt(x)))/((1+x^{1/4))}

Lösung

integral

\frac{( x )}{( 1 + x ^{\frac{1}{4}} )}

Lösung

\frac{4}{5} x^{\frac{5}{4}} - x + \frac{4}{3} x^{\frac{3}{4}} - 2 x + 4 4 x - 4 ln 4 x + 1 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{4 u ^{5}}{1 + u} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{u ^{5}}{1 + u} d u

Lange Division

\frac{u ^{5}}{1 + u} : u^{4} - u^{3} + u^{2} - u + 1 - \frac{1}{u + 1}

= 4 \cdot \int u^{4} - u^{3} + u^{2} - u + 1 - \frac{1}{u + 1} 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

= 4 (\int u^{4} d u - \int u^{3} d u + \int u^{2} d u - \int u d u + \int 1 d u - \int \frac{1}{u + 1} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{3} d u = \frac{u ^{4}}{4}

\int u^{2} d u = \frac{u ^{3}}{3}

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

\int 1 d u = u

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= 4 (\frac{u ^{5}}{5} - \frac{u ^{4}}{4} + \frac{u ^{3}}{3} - \frac{u ^{2}}{2} + u - ln ∣ u + 1 ∣)

Setze in

u = x^{\frac{1}{4}}

ein

= 4 \frac{( x ^{\frac{1}{4}} ) ^{5}}{5} - \frac{( x ^{\frac{1}{4}} ) ^{4}}{4} + \frac{( x ^{\frac{1}{4}} ) ^{3}}{3} - \frac{( x ^{\frac{1}{4}} ) ^{2}}{2} + x^{\frac{1}{4}} - ln x^{\frac{1}{4}} + 1

Vereinfache

4 \frac{( x ^{\frac{1}{4}} ) ^{5}}{5} - \frac{( x ^{\frac{1}{4}} ) ^{4}}{4} + \frac{( x ^{\frac{1}{4}} ) ^{3}}{3} - \frac{( x ^{\frac{1}{4}} ) ^{2}}{2} + x^{\frac{1}{4}} - ln x^{\frac{1}{4}} + 1 : \frac{4}{5} x^{\frac{5}{4}} - x + \frac{4}{3} x^{\frac{3}{4}} - 2 x + 4 4 x - 4 ln 4 x + 1

= \frac{4}{5} x^{\frac{5}{4}} - x + \frac{4}{3} x^{\frac{3}{4}} - 2 x + 4 4 x - 4 ln 4 x + 1

Füge eine Konstante zur Lösung hinzu

= \frac{4}{5} x^{\frac{5}{4}} - x + \frac{4}{3} x^{\frac{3}{4}} - 2 x + 4 4 x - 4 ln 4 x + 1 + C

Graph

Beliebte Beispiele

integral (x^2)/9

in t e g r a l \frac{x ^{2}}{9}

integral (e^{-x})/2

in t e g r a l \frac{e ^{- x}}{2}

integral 6/((1+t^2))

in t e g r a l \frac{6}{( 1 + t ^{2} )}

integral sqrt(a^2+x^2)

in t e g r a l a^{2} + x^{2}

integral ln(1+x^2)

in t e g r a l ln (1 + x^{2})