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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{17} (6 ln x^{2} + 1 + 3 arctan (x)) - \frac{7}{34} ln ∣ 4 x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{2 x ^{2} - 1}{( x ^{2} + 1 ) ( 4 x - 1 )} : \frac{12 x + 3}{17 ( x ^{2} + 1 )} - \frac{14}{17 ( 4 x - 1 )}

= \int \frac{12 x + 3}{17 ( x ^{2} + 1 )} - \frac{14}{17 ( 4 x - 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{12 x + 3}{17 ( x ^{2} + 1 )} d x - \int \frac{14}{17 ( 4 x - 1 )} d x

\int \frac{12 x + 3}{17 ( x ^{2} + 1 )} d x = \frac{1}{17} (6 ln x^{2} + 1 + 3 arctan (x))

\int \frac{14}{17 ( 4 x - 1 )} d x = \frac{7}{34} ln ∣ 4 x - 1 ∣

= \frac{1}{17} (6 ln x^{2} + 1 + 3 arctan (x)) - \frac{7}{34} ln ∣ 4 x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{17} (6 ln x^{2} + 1 + 3 arctan (x)) - \frac{7}{34} ln ∣ 4 x - 1 ∣ + C

Graph

Beliebte Beispiele

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

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

integral of sin^3(t)cos^3(t)

\int sin^{3} (t) cos^{3} (t) d t

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

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

integral of (1+3x)

\int (1 + 3 x) d x

integral of (4x^2-5)^{1/2}

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