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Beliebt Rechnen >

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

Lösung

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

Lösung

3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u - 5 ) ^{2}}{u} : u - 10 + \frac{25}{u}

= 3 \cdot \int u - 10 + \frac{25}{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

= 3 (\int u d u - \int 10 d u + \int \frac{25}{u} d u)

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

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= 3 (\frac{u ^{2}}{2} - 10 u + 25 ln ∣ u ∣)

Setze in

u = r + 5

ein

= 3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣)

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣) + C

Graph

Beliebte Beispiele

slopeintercept (3,5),(1,-7)

s l o p e in t erce pt (3, 5), (1, - 7)

limit as x approaches 0 of ((2+x^3)-8)/x

x \to 0 lim (\frac{( 2 + x ^{3} ) - 8}{x})

integral from 0 to 2 of (5t)/((t-3)^2)

\int_{0}^{2} \frac{5 t}{( t - 3 ) ^{2}} d t

derivative of ln(-5x)

\frac{d}{d x} (ln (- 5 x))

inverselaplace (1/3)/s

in v erse l a pl a ce \frac{\frac{1}{3}}{s}