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Beliebt >

integral of 1/(sqrt((x^2-6x+10)^5))

Lösung

\int \frac{1}{( x ^{2} - 6 x + 10 ) ^{5}} d x

\frac{x - 3}{x ^{2} - 6 x + 10} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} - 6 x + 10 ) ^{5}} d x

Vervollständige das Quadrat

x^{2} - 6 x + 10 : (x - 3)^{2} + 1

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

Wende U-Substitution an

: \int \frac{1}{( u ^{2} + 1 ) ^{\frac{5}{2}}} d u

= \int \frac{1}{( u ^{2} + 1 ) ^{\frac{5}{2}}} d u

Trigonometrische Substitution anwenden

: \int cos^{3} (v) d v

= \int cos^{3} (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (1 - sin^{2} (v)) cos (v) d v

Wende U-Substitution an

: \int 1 - w^{2} d w

= \int 1 - w^{2} d w

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d w - \int w^{2} d w

\int 1 d w = w

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

= w - \frac{w ^{3}}{3}

Ersetze zurück

= sin (arctan (x - 3)) - \frac{sin ^{3} ( arctan ( x - 3 ) )}{3}

Vereinfache

sin (arctan (x - 3)) - \frac{sin ^{3} ( arctan ( x - 3 ) )}{3} : \frac{x - 3}{x ^{2} - 6 x + 10} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}}

= \frac{x - 3}{x ^{2} - 6 x + 10} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}}

Füge eine Konstante zur Lösung hinzu

= \frac{x - 3}{x ^{2} - 6 x + 10} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}} + C

\int \frac{x ^{2}}{4 - x ^{2}} d x

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

\int \frac{4 - x ^{2}}{x} d x

\int e^{9 t} d t

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