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Beliebt Rechnen >

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

Lösung

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

Lösung

x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2} + C

Schritte zur Lösung

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

Faktorisiere

\frac{2 x ^{2} + 1}{2 x ^{2} + x}

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

Multipliziere aus

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

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

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

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

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

Vereinfache

x + \frac{1}{2} - \frac{1}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ : x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2}

= x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2}

Füge eine Konstante zur Lösung hinzu

= x - \frac{3}{2} ln ∣ 2 x + 1 ∣ + ln ∣ x ∣ + \frac{1}{2} + C

Graph

Beliebte Beispiele

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\int x^{3} 9 + x^{4} d x

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

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

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\int \frac{( a + b x ^{5} )}{6 a x + b x ^{6}} d x

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\int \frac{7}{tan ( t ) ( csc ( t ) + 4 sin ( t ) )} d t