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Beliebt Rechnen >

integral of (8x^2-4x+7)/(4x^3+x^2+4x+1)

Lösung

\int \frac{8 x ^{2} - 4 x + 7}{4 x ^{3} + x ^{2} + 4 x + 1} d x

Lösung

2 ln ∣ 4 x + 1 ∣ - arctan (x) + C

Schritte zur Lösung

\int \frac{8 x ^{2} - 4 x + 7}{4 x ^{3} + x ^{2} + 4 x + 1} d x

Ermittle den Partialbruch von

\frac{8 x ^{2} - 4 x + 7}{4 x ^{3} + x ^{2} + 4 x + 1} : \frac{8}{4 x + 1} - \frac{1}{x ^{2} + 1}

= \int \frac{8}{4 x + 1} - \frac{1}{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{8}{4 x + 1} d x - \int \frac{1}{x ^{2} + 1} d x

\int \frac{8}{4 x + 1} d x = 2 ln ∣ 4 x + 1 ∣

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

= 2 ln ∣ 4 x + 1 ∣ - arctan (x)

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ 4 x + 1 ∣ - arctan (x) + C

Graph

Beliebte Beispiele

tangent f(x)=-x^2-4x+2,\at x=-1

t an g e n t f (x) = - x^{2} - 4 x + 2, at x = - 1

derivative of (x^2-36/(x+6))

\frac{d}{d x} (\frac{x ^{2} - 36}{x + 6})

derivative of 3^{x+10}

\frac{d}{d x} (3^{x + 10})

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

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

(2y+2t-7)dt+(4y+2t-1)dy=0

(2 y + 2 t - 7) d t + (4 y + 2 t - 1) d y = 0