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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

\frac{1}{x} 1 + \frac{1}{x ^{4}} : \frac{x ^{4} + 1}{x x ^{4}}

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

x^{4} = x^{2},

angenommen

x \geq 0

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

x x^{2} = x^{3}

= \int \frac{x ^{4} + 1}{x ^{3}} d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

y^{''}+4y^'+3y=32xe^x

y^{^{''}} + 4 y^{^{'}} + 3 y = 32 x e^{x}

derivative of 3x(18x^4+(13/(x+1)))

\frac{d}{d x} (3 x (18 x^{4} + \frac{13}{x + 1}))

derivative of ln(sin(x^3))

\frac{d}{d x} (ln (sin (x^{3})))

tangent xy=-121

t an g e n t x y = - 121

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