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Beliebt Rechnen >

integral of ((4x-5))/((x^2-2x+2)^{3/2)}

Lösung

\int \frac{( 4 x - 5 )}{( x ^{2} - 2 x + 2 ) ^{\frac{3}{2}}} d x

Lösung

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

Schritte zur Lösung

\int \frac{4 x - 5}{( x ^{2} - 2 x + 2 ) ^{\frac{3}{2}}} d x

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

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

Setze in

u = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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\int \frac{1}{( x ^{2} + 49 ) ^{3}} d x

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