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

Lösung

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

13 ln ∣ x ∣ - 13 ln ∣ x + 1 ∣ + \frac{7}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{6}{x} - \frac{1}{x ^{2}} + \frac{1}{( x + 1 ) ^{2}} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{2}{x ^{3} ( x + 1 ) ^{3}} d x = 12 ln ∣ x ∣ + \frac{6}{x} - \frac{1}{x ^{2}} - 12 ln ∣ x + 1 ∣ + \frac{6}{x + 1} + \frac{1}{( x + 1 ) ^{2}}

= ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + 12 ln ∣ x ∣ + \frac{6}{x} - \frac{1}{x ^{2}} - 12 ln ∣ x + 1 ∣ + \frac{6}{x + 1} + \frac{1}{( x + 1 ) ^{2}}

Vereinfache

ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + 12 ln ∣ x ∣ + \frac{6}{x} - \frac{1}{x ^{2}} - 12 ln ∣ x + 1 ∣ + \frac{6}{x + 1} + \frac{1}{( x + 1 ) ^{2}} : 13 ln ∣ x ∣ - 13 ln ∣ x + 1 ∣ + \frac{7}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{6}{x} - \frac{1}{x ^{2}} + \frac{1}{( x + 1 ) ^{2}}

= 13 ln ∣ x ∣ - 13 ln ∣ x + 1 ∣ + \frac{7}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{6}{x} - \frac{1}{x ^{2}} + \frac{1}{( x + 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= 13 ln ∣ x ∣ - 13 ln ∣ x + 1 ∣ + \frac{7}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{6}{x} - \frac{1}{x ^{2}} + \frac{1}{( x + 1 ) ^{2}} + C

a re a \frac{1}{( 1 - 7 x )}, - 6 < x < 0

\frac{\partial}{\partial x} (4 x^{- x^{2} - y^{2} - z^{2}})

\int 7.9 t - 16.6 d t

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

\frac{\partial}{\partial x} (x^{\frac{2}{3}} + y^{\frac{2}{3}})