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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x^3ln(sqrt(x)))/3

\int \frac{x ^{3} ln ( x )}{3} d x

integral of x^{-1/2}+x^{3/2}

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

integral of (-x^3+6x^2-9x)

\int (- x^{3} + 6 x^{2} - 9 x) d x

integral of ((tan(ln(x)))/x)

\int (\frac{tan ( ln ( x ) )}{x}) d x

integral of (4x+10)/((x^2+5x+12)^2)

\int \frac{4 x + 10}{( x ^{2} + 5 x + 12 ) ^{2}} d x