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Beliebt Rechnen >

integral of (7x^2-4x+2)/(x(x-4)^2)

Lösung

\int \frac{7 x ^{2} - 4 x + 2}{x ( x - 4 ) ^{2}} d x

Lösung

\frac{1}{8} ln ∣ x ∣ + \frac{55}{8} ln ∣ x - 4 ∣ - \frac{24}{x - 4} - \frac{1}{2 ( x - 4 )} + C

Schritte zur Lösung

\int \frac{7 x ^{2} - 4 x + 2}{x ( x - 4 ) ^{2}} d x

Multipliziere aus

\frac{7 x ^{2} - 4 x + 2}{x ( x - 4 ) ^{2}} : \frac{7 x ^{2}}{x ( x - 4 ) ^{2}} - \frac{4 x}{x ( x - 4 ) ^{2}} + \frac{2}{x ( x - 4 ) ^{2}}

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

\int \frac{7 x ^{2}}{x ( x - 4 ) ^{2}} d x = 7 (ln ∣ x - 4 ∣ - \frac{4}{x - 4})

\int \frac{4 x}{x ( x - 4 ) ^{2}} d x = - \frac{4}{x - 4}

\int \frac{2}{x ( x - 4 ) ^{2}} d x = 2 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )})

= 7 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) - (- \frac{4}{x - 4}) + 2 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )})

Vereinfache

7 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) - (- \frac{4}{x - 4}) + 2 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )}) : \frac{1}{8} ln ∣ x ∣ + \frac{55}{8} ln ∣ x - 4 ∣ - \frac{24}{x - 4} - \frac{1}{2 ( x - 4 )}

= \frac{1}{8} ln ∣ x ∣ + \frac{55}{8} ln ∣ x - 4 ∣ - \frac{24}{x - 4} - \frac{1}{2 ( x - 4 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} ln ∣ x ∣ + \frac{55}{8} ln ∣ x - 4 ∣ - \frac{24}{x - 4} - \frac{1}{2 ( x - 4 )} + C

Graph

Beliebte Beispiele

integral of (-x^2+8)/(x^3+4x)

\int \frac{- x ^{2} + 8}{x ^{3} + 4 x} d x

integral of (2x+4y)

\int (2 x + 4 y) d x

integral of (2e^x+sin(x)-cos(x))

\int (2 e^{x} + sin (x) - cos (x)) d x

integral of 1/(x(x^6+8))

\int \frac{1}{x ( x ^{6} + 8 )} d x

integral of x/(2+4x^2)

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