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Beliebt Rechnen >

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

Lösung

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

Lösung

4 x - 5 arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= 4 (- arctan (x) + x) - arctan (x)

Vereinfache

4 (- arctan (x) + x) - arctan (x) : 4 x - 5 arctan (x)

= 4 x - 5 arctan (x)

Füge eine Konstante zur Lösung hinzu

= 4 x - 5 arctan (x) + C

Graph

Beliebte Beispiele

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

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

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

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