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Beliebt Rechnen >

integral of (x^2-x+1)/(sqrt((x^2+1)^3))

Lösung

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

Lösung

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

Schritte zur Lösung

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

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

angenommen

(x^{2} + 1) \geq 0

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

Multipliziere aus

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

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

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

\int \frac{x ^{2}}{( x ^{2} + 1 ) ^{\frac{3}{2}}} d x = ln x + 1 + x^{2} - \frac{x}{1 + x ^{2}}

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative 3e^x+8/(\sqrt[3]{x)}

d er i v a t i v e 3 e^{x} + \frac{8}{3 x}

derivative of ln(sqrt(pi))

\frac{d}{d x} (ln (π))

integral of (x^3-x^2)/(x^2)

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

derivative of cos(2x^3-x)

\frac{d}{d x} (cos (2 x^{3} - x))

derivative f(x)=ln(x+2)

d er i v a t i v e f (x) = ln (x + 2)