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Beliebt Rechnen >

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

Lösung

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

Lösung

2 (- 2 arctan (\frac{1}{2} (x - 2)) + x - 2 + \frac{3}{2} arctan (\frac{1}{2} (x - 2))) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2 ( u ^{2} + 3 )}{u ^{2} + 2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{u ^{2} + 3}{u ^{2} + 2} d u

Multipliziere aus

\frac{u ^{2} + 3}{u ^{2} + 2} : \frac{u ^{2}}{u ^{2} + 2} + \frac{3}{u ^{2} + 2}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{u ^{2}}{u ^{2} + 2} d u + \int \frac{3}{u ^{2} + 2} d u)

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

\int \frac{3}{u ^{2} + 2} d u = \frac{3}{2} arctan (\frac{u}{2})

= 2 (- 2 arctan (\frac{u}{2}) + u + \frac{3}{2} arctan (\frac{u}{2}))

Setze in

u = x - 2

ein

= 2 (- 2 arctan (\frac{x - 2}{2}) + x - 2 + \frac{3}{2} arctan (\frac{x - 2}{2}))

Vereinfache

2 (- 2 arctan (\frac{x - 2}{2}) + x - 2 + \frac{3}{2} arctan (\frac{x - 2}{2})) : 2 (- 2 arctan (\frac{1}{2} (x - 2)) + x - 2 + \frac{3}{2} arctan (\frac{1}{2} (x - 2)))

= 2 (- 2 arctan (\frac{1}{2} (x - 2)) + x - 2 + \frac{3}{2} arctan (\frac{1}{2} (x - 2)))

Füge eine Konstante zur Lösung hinzu

= 2 (- 2 arctan (\frac{1}{2} (x - 2)) + x - 2 + \frac{3}{2} arctan (\frac{1}{2} (x - 2))) + C

Graph

Beliebte Beispiele

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

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

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

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

integral of 1/(sin(x)-tan(x))

\int \frac{1}{sin ( x ) - tan ( x )} d x

integral of 3x^2(x^3+4)^5

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

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

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