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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x + 1 ∣ - \frac{1}{2} ln x^{2} + 1 + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 1 ∣ - \frac{1}{2} ln x^{2} + 1 + C

Graph

Beliebte Beispiele

integral of (5e^{-3+ln(3x)})/x

\int \frac{5 e ^{- 3 + l n (3 x)}}{x} d x

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

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

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

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

integral of 6/(x^3+4x^2+6x)

\int \frac{6}{x ^{3} + 4 x ^{2} + 6 x} d x

integral of 18x^2\sqrt[4]{6x^3+5}

\int 18 x^{2} 4 6 x^{3} + 5 d x