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Beliebt Rechnen >

integral of (x^2)/((x-3)^5)

Lösung

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

Lösung

- \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{( u + 3 ) ^{2}}{u ^{5}} d u

Multipliziere aus

\frac{( u + 3 ) ^{2}}{u ^{5}} : \frac{1}{u ^{3}} + \frac{6}{u ^{4}} + \frac{9}{u ^{5}}

= \int \frac{1}{u ^{3}} + \frac{6}{u ^{4}} + \frac{9}{u ^{5}} 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{1}{u ^{3}} d u + \int \frac{6}{u ^{4}} d u + \int \frac{9}{u ^{5}} d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{6}{u ^{4}} d u = - \frac{2}{u ^{3}}

\int \frac{9}{u ^{5}} d u = - \frac{9}{4 u ^{4}}

= - \frac{1}{2 u ^{2}} - \frac{2}{u ^{3}} - \frac{9}{4 u ^{4}}

Setze in

u = x - 3

ein

= - \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{3}} - \frac{9}{4 ( x - 3 ) ^{4}} + C

Graph

Beliebte Beispiele

integral of 3x(x^2+8)^2

\int 3 x (x^{2} + 8)^{2} d x

integral of x^2(x^3+8)^{10}

\int x^{2} (x^{3} + 8)^{10} d x

integral of 1/((x+3)sqrt(x+1))

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

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

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

integral of (e^{3x})/(3-e^{3x)}

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