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Beliebt Rechnen >

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

Lösung

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

Lösung

2 x - 3 arctan (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= 2 (- arctan (x) + x) - arctan (x)

Vereinfache

2 (- arctan (x) + x) - arctan (x) : 2 x - 3 arctan (x)

= 2 x - 3 arctan (x)

Füge eine Konstante zur Lösung hinzu

= 2 x - 3 arctan (x) + C

Graph

Beliebte Beispiele

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

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

integral of (sqrt(x^2-5))/(x^2)

\int \frac{x ^{2} - 5}{x ^{2}} d x

integral of (sec(2θ)-csc(θ/2))

\int (sec (2 θ) - csc (\frac{θ}{2})) d θ

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

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

integral of 2sin(2θ)sin(4θ)

\int 2 sin (2 θ) sin (4 θ) d θ