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Beliebt Rechnen >

integral of (x+3)/((x-2)(x+5))

Lösung

\int \frac{x + 3}{( x - 2 ) ( x + 5 )} d x

Lösung

ln ∣ x + 5 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣) + C

Schritte zur Lösung

\int \frac{x + 3}{( x - 2 ) ( x + 5 )} d x

Wende U-Substitution an

= \int \frac{u + 5}{u ( u + 7 )} d u

Multipliziere aus

\frac{u + 5}{u ( u + 7 )} : \frac{u}{u ( u + 7 )} + \frac{5}{u ( u + 7 )}

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{u}{u ( u + 7 )} d u + \int \frac{5}{u ( u + 7 )} d u

\int \frac{u}{u ( u + 7 )} d u = ln ∣ u + 7 ∣

\int \frac{5}{u ( u + 7 )} d u = 5 (\frac{1}{7} ln ∣ u ∣ - \frac{1}{7} ln ∣ u + 7 ∣)

= ln ∣ u + 7 ∣ + 5 (\frac{1}{7} ln ∣ u ∣ - \frac{1}{7} ln ∣ u + 7 ∣)

Setze in

u = x - 2

ein

= ln ∣ x - 2 + 7 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x - 2 + 7 ∣)

Vereinfache

ln ∣ x - 2 + 7 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x - 2 + 7 ∣) : ln ∣ x + 5 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣)

= ln ∣ x + 5 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 5 ∣ + 5 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣) + C

Graph

Beliebte Beispiele

(\partial)/(\partial x)((x-1)^{2/3}(y^2-4))

\frac{\partial}{\partial x} ((x - 1)^{\frac{2}{3}} (y^{2} - 4))

(\partial)/(\partial y)(y(x+y+1))

\frac{\partial}{\partial y} (y (x + y + 1))

derivative of (x-5^3)

\frac{d}{d x} ((x - 5)^{3})

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

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

integral of 4/(1-x^2)

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