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Beliebt Rechnen >

integral of (6x^2+5x+4)/((1+x)(1+4x^2))

Lösung

\int \frac{6 x ^{2} + 5 x + 4}{( 1 + x ) ( 1 + 4 x ^{2} )} d x

Lösung

ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} + 1 + \frac{3}{2} arctan (2 x) + C

Schritte zur Lösung

\int \frac{6 x ^{2} + 5 x + 4}{( 1 + x ) ( 1 + 4 x ^{2} )} d x

Ermittle den Partialbruch von

\frac{6 x ^{2} + 5 x + 4}{( 1 + x ) ( 1 + 4 x ^{2} )} : \frac{1}{x + 1} + \frac{2 x + 3}{4 x ^{2} + 1}

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

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

\int \frac{2 x + 3}{4 x ^{2} + 1} d x = \frac{1}{4} ln 4 x^{2} + 1 + \frac{3}{2} arctan (2 x)

= ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} + 1 + \frac{3}{2} arctan (2 x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} + 1 + \frac{3}{2} arctan (2 x) + C

Graph

Beliebte Beispiele

integral of (2x^3+5x)/(x^4+5x^2+4)

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

integral of x^2y^3-1/(1+9x^2)

\int x^{2} y^{3} - \frac{1}{1 + 9 x ^{2}} d x

integral of (csc(x)+cot(x))^2

\int (csc (x) + cot (x))^{2} d x

integral of za^z

\int z a^{z} d z

integral of (sqrt(x^2+2x-3))/((x+1)^4)

\int \frac{x ^{2} + 2 x - 3}{( x + 1 ) ^{4}} d x