integral of (5r^2)/(r+3)

Lösung

\int \frac{5 r ^{2}}{r + 3} d r

5 (\frac{( r + 3 ) ^{2}}{2} - 6 (r + 3) + 9 ln ∣ r + 3 ∣) + C

Schritte zur Lösung

\int \frac{5 r ^{2}}{r + 3} d r

Entferne die Konstante:

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

= 5 \cdot \int \frac{r ^{2}}{r + 3} d r

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 3 ) ^{2}}{u} : u - 6 + \frac{9}{u}

= 5 \cdot \int u - 6 + \frac{9}{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

= 5 (\int u d u - \int 6 d u + \int \frac{9}{u} d u)

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

\int 6 d u = 6 u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

= 5 (\frac{u ^{2}}{2} - 6 u + 9 ln ∣ u ∣)

Setze in

u = r + 3

ein

= 5 (\frac{( r + 3 ) ^{2}}{2} - 6 (r + 3) + 9 ln ∣ r + 3 ∣)

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{( r + 3 ) ^{2}}{2} - 6 (r + 3) + 9 ln ∣ r + 3 ∣) + C

\frac{d}{d x} (x^{2} ln (5 x))

d er i v a t i v e (x - 2)^{2}

f a c t or 7 p^{3} + 4 p

d er i v a t i v e 2 e^{3 x}

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