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Beliebt Rechnen >

integral of sqrt(1/2+(x^4)/4+1/(4x^4))

Lösung

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

Lösung

\frac{1}{2} (\frac{x ^{3}}{3} - \frac{1}{x}) + C

Schritte zur Lösung

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

Vereinfache

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

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

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{( x ^{2} + 2 x + 1 ) ^{2} ( x ^{2} - 2 x + 1 ) ^{2}}{x ^{4}} d x

Vereinfache

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

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

Multipliziere aus

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

= \frac{1}{2} \cdot \int x^{2} + \frac{1}{x ^{2}} d x

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

y^{''}+4y^'+4y=12t+16

y^{^{''}} + 4 y^{^{'}} + 4 y = 12 t + 16

integral from 0 to 10 of sqrt(1+16x^2)

\int_{0}^{10} 1 + 16 x^{2} d x

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

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

limit as x approaches 5 of 2/(5-x)

x \to 5 lim (\frac{2}{5 - x})

derivative of (1+8/x ^x)

\frac{d}{d x} ((1 + \frac{8}{x})^{x})