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Beliebt Rechnen >

integral of (4x^2+10x+13)/((x+2)(x^2+5))

Lösung

\int \frac{4 x ^{2} + 10 x + 13}{( x + 2 ) ( x ^{2} + 5 )} d x

Lösung

ln ∣ x + 2 ∣ + \frac{3}{2} ln x^{2} + 5 + \frac{4}{5} arctan (\frac{x}{5}) + C

Schritte zur Lösung

\int \frac{4 x ^{2} + 10 x + 13}{( x + 2 ) ( x ^{2} + 5 )} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} + 10 x + 13}{( x + 2 ) ( x ^{2} + 5 )} : \frac{1}{x + 2} + \frac{3 x + 4}{x ^{2} + 5}

= \int \frac{1}{x + 2} + \frac{3 x + 4}{x ^{2} + 5} 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{3 x + 4}{x ^{2} + 5} d x

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

\int \frac{3 x + 4}{x ^{2} + 5} d x = \frac{3}{2} ln x^{2} + 5 + \frac{4}{5} arctan (\frac{x}{5})

= ln ∣ x + 2 ∣ + \frac{3}{2} ln x^{2} + 5 + \frac{4}{5} arctan (\frac{x}{5})

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 2 ∣ + \frac{3}{2} ln x^{2} + 5 + \frac{4}{5} arctan (\frac{x}{5}) + C

Graph

Beliebte Beispiele

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

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

integral of 9/(xln(5x))

\int \frac{9}{x ln ( 5 x )} d x

integral of 1/((t-2)^2+9)

\int \frac{1}{( t - 2 ) ^{2} + 9} d t

integral of (x-3)sqrt(2x+1)

\int (x - 3) 2 x + 1 d x

integral of 4xsqrt(x-2)

\int 4 x x - 2 d x