integral of (x^3)/(x^2+10x+25)

Lösung

\int \frac{x ^{3}}{x ^{2} + 10 x + 25} d x

\frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5} + C

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 10 x + 25} d x

Vervollständige das Quadrat

x^{2} + 10 x + 25 : (x + 5)^{2}

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

Wende U-Substitution an

= \int \frac{( u - 5 ) ^{3}}{u ^{2}} d u

Multipliziere aus

\frac{( u - 5 ) ^{3}}{u ^{2}} : u - 15 + \frac{75}{u} - \frac{125}{u ^{2}}

= \int u - 15 + \frac{75}{u} - \frac{125}{u ^{2}} d u

Wende die Summenregel an:

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

= \int u d u - \int 15 d u + \int \frac{75}{u} d u - \int \frac{125}{u ^{2}} d u

\int u d u = \frac{u ^{2}}{2}

\int 15 d u = 15 u

\int \frac{75}{u} d u = 75 ln ∣ u ∣

\int \frac{125}{u ^{2}} d u = - \frac{125}{u}

= \frac{u ^{2}}{2} - 15 u + 75 ln ∣ u ∣ - (- \frac{125}{u})

Setze in

u = x + 5

ein

= \frac{( x + 5 ) ^{2}}{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ - (- \frac{125}{x + 5})

Vereinfache

\frac{( x + 5 ) ^{2}}{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ - (- \frac{125}{x + 5}) : \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5}

= \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5} + C

\frac{\partial}{\partial x} (lo g_{10} (x + y))

\int t sin (9 t) d t

s l o p e (2.3) (4.4)

\frac{\partial}{\partial y} (4 y x)

x \to 3 lim (\frac{2}{x ^{2} - 9})