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Beliebt Rechnen >

integral of (x^2-2x+1)/(4(x-1))

Lösung

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

Lösung

\frac{1}{8} (x^{2} - 2 x + 1) + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{x ^{2} - 2 x + 1}{x - 1} d x

Multipliziere aus

\frac{x ^{2} - 2 x + 1}{x - 1} : \frac{x ^{2}}{x - 1} - \frac{2 x}{x - 1} + \frac{1}{x - 1}

Wende die Summenregel an:

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

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

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

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

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

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

Vereinfache

\frac{1}{4} (\frac{( x - 1 ) ^{2}}{2} + 2 (x - 1) + ln ∣ x - 1 ∣ - 2 (x - 1 + ln ∣ x - 1 ∣) + ln ∣ x - 1 ∣) : \frac{1}{8} (x^{2} - 2 x + 1)

= \frac{1}{8} (x^{2} - 2 x + 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (x^{2} - 2 x + 1) + C

Graph

Beliebte Beispiele

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

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

integral of (e^{arctan(x)}-2x)/(1+x^2)

\int \frac{e ^{a r c t a n (x)} - 2 x}{1 + x ^{2}} d x

integral of (x+4)e^{-2x}

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

integral of (16x^2)/(sqrt(49-x^2))

\int \frac{16 x ^{2}}{49 - x ^{2}} d x

integral of 1-1(2x+2)

\int 1 - 1 (2 x + 2) d x