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Beliebt Rechnen >

integral of (2x-3)/(x^2+2x-17)

Lösung

\int \frac{2 x - 3}{x ^{2} + 2 x - 17} d x

Lösung

ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 + \frac{1}{2 2} (ln \frac{1}{3 2} (x + 1) + 1 - ln \frac{1}{3 2} (x + 1) - 1) + C

Schritte zur Lösung

\int \frac{2 x - 3}{x ^{2} + 2 x - 17} d x

Multipliziere aus

\frac{2 x - 3}{x ^{2} + 2 x - 17} : \frac{2 x}{x ^{2} + 2 x - 17} - \frac{3}{x ^{2} + 2 x - 17}

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}{x ^{2} + 2 x - 17} d x - \int \frac{3}{x ^{2} + 2 x - 17} d x

\int \frac{2 x}{x ^{2} + 2 x - 17} d x = ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1

\int \frac{3}{x ^{2} + 2 x - 17} d x = - \frac{1}{2 2} (ln \frac{x + 1}{3 2} + 1 - ln \frac{x + 1}{3 2} - 1)

= ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 - (- \frac{1}{2 2} (ln \frac{x + 1}{3 2} + 1 - ln \frac{x + 1}{3 2} - 1))

Vereinfache

ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 - (- \frac{1}{2 2} (ln \frac{x + 1}{3 2} + 1 - ln \frac{x + 1}{3 2} - 1)) : ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 + \frac{1}{2 2} (ln \frac{1}{3 2} (x + 1) + 1 - ln \frac{1}{3 2} (x + 1) - 1)

= ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 + \frac{1}{2 2} (ln \frac{1}{3 2} (x + 1) + 1 - ln \frac{1}{3 2} (x + 1) - 1)

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 2 x - 17 + \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) + 1 - \frac{1}{3 2} ln \frac{1}{3 2} (x + 1) - 1 + \frac{1}{2 2} (ln \frac{1}{3 2} (x + 1) + 1 - ln \frac{1}{3 2} (x + 1) - 1) + C

Graph

Beliebte Beispiele

y^{1/2}(dy)/(dx)+y^{3/2}=1,y(0)=16

y^{\frac{1}{2}} \frac{d y}{d x} + y^{\frac{3}{2}} = 1, y (0) = 16

integral of (sin(2θ))/(1+sin^2(θ))

\int \frac{sin ( 2 θ )}{1 + sin ^{2} ( θ )} d θ

integral of sqrt(2+4x^2)

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

integral of e^xsqrt(1-e^x)

\int e^{x} 1 - e^{x} d x

implicit e^{x^2y}=x+y

im pl i c i t e^{x^{2} y} = x + y