integral of (y^2)/(y+1)

Lösung

\int \frac{y ^{2}}{y + 1} d y

\frac{( y + 1 ) ^{2}}{2} - 2 (y + 1) + ln ∣ y + 1 ∣ + C

Schritte zur Lösung

\int \frac{y ^{2}}{y + 1} d y

Wende U-Substitution an

= \int \frac{( u - 1 ) ^{2}}{u} d u

Multipliziere aus

\frac{( u - 1 ) ^{2}}{u} : u - 2 + \frac{1}{u}

= \int u - 2 + \frac{1}{u} 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 2 d u + \int \frac{1}{u} d u

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + ln ∣ u ∣

Setze in

u = y + 1

ein

= \frac{( y + 1 ) ^{2}}{2} - 2 (y + 1) + ln ∣ y + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( y + 1 ) ^{2}}{2} - 2 (y + 1) + ln ∣ y + 1 ∣ + C

\frac{d}{d x} (2 sec (x))

\frac{d}{d x} (\frac{3 - 8 x}{x + 4})

in v erse l a pl a ce \frac{s ^{3} + s + 6}{s ^{4} ( s + 2 )}

\int_{7}^{8} \frac{x}{x ^{2} - 3} d x

x \to 7 + lim (3 x + 5)