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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{2} ln 16 x^{2} - 8 x + 20 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{- 4 x + 1}{4 x ^{2} - 2 x + 5} : - \frac{4 x}{4 x ^{2} - 2 x + 5} + \frac{1}{4 x ^{2} - 2 x + 5}

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

\int \frac{4 x}{4 x ^{2} - 2 x + 5} d x = \frac{1}{2} ln 16 x^{2} - 8 x + 20 + \frac{1}{19} arctan (\frac{4 x - 1}{19})

\int \frac{1}{4 x ^{2} - 2 x + 5} d x = \frac{1}{19} arctan (\frac{4 x - 1}{19})

= - (\frac{1}{2} ln 16 x^{2} - 8 x + 20 + \frac{1}{19} arctan (\frac{4 x - 1}{19})) + \frac{1}{19} arctan (\frac{4 x - 1}{19})

Vereinfache

- (\frac{1}{2} ln 16 x^{2} - 8 x + 20 + \frac{1}{19} arctan (\frac{4 x - 1}{19})) + \frac{1}{19} arctan (\frac{4 x - 1}{19}) : - \frac{1}{2} ln 16 x^{2} - 8 x + 20

= - \frac{1}{2} ln 16 x^{2} - 8 x + 20

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln 16 x^{2} - 8 x + 20 + C

Graph

Beliebte Beispiele

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

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

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\int 7 x^{\frac{- 3}{4}} d x

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\int 8 tan (x) sec^{5} (x) d x

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\int \frac{( 16 - t ^{2} ) ^{\frac{3}{2}}}{t ^{6}} d t

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