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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

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

integral of x^2arctan(2x^3)

\int x^{2} arctan (2 x^{3}) d x

integral of 1/(xsqrt(x^2+36))

\int \frac{1}{x x ^{2} + 36} d x

integral of 1/(uln(u))

\int \frac{1}{u ln ( u )} d u

integral of 3^{t^2}

\int 3^{t^{2}} d t

derivative of (3+2x/(sqrt(x)))

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