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integral of (sqrt(x^2+6x+18))/(x+3)

Lösung

\int \frac{x ^{2} + 6 x + 18}{x + 3} d x

- \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} + 1 + \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} - 1 + x^{2} + 6 x + 18 + C

Schritte zur Lösung

\int \frac{x ^{2} + 6 x + 18}{x + 3} d x

Wende U-Substitution an

= \int \frac{u ^{2} + 9}{u} d u

Wende U-Substitution an

= \int \frac{v ^{2}}{v ^{2} - 9} d v

Faktorisiere

v^{2} - 9 : - (- v^{2} + 9)

= \int \frac{v ^{2}}{- ( - v ^{2} + 9 )} d v

Entferne die Konstante:

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

= - \int \frac{v ^{2}}{- v ^{2} + 9} d v

Wende integrale Substitution an

= - \int \frac{3 w ^{2}}{- w ^{2} + 1} d w

Entferne die Konstante:

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

= - 3 \cdot \int \frac{w ^{2}}{- w ^{2} + 1} d w

\frac{w ^{2}}{- w ^{2} + 1} = \frac{1}{- w ^{2} + 1} - 1

= - 3 \cdot \int \frac{1}{- w ^{2} + 1} - 1 d w

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

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

\int 1 d w = w

= - 3 (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2} - w)

Ersetze zurück

= - 3 \frac{ln \frac{( x + 3 ) ^{2} + 9}{3} + 1}{2} - \frac{ln \frac{( x + 3 ) ^{2} + 9}{3} - 1}{2} - \frac{( x + 3 ) ^{2} + 9}{3}

Vereinfache

- 3 \frac{ln \frac{( x + 3 ) ^{2} + 9}{3} + 1}{2} - \frac{ln \frac{( x + 3 ) ^{2} + 9}{3} - 1}{2} - \frac{( x + 3 ) ^{2} + 9}{3} : - \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} + 1 + \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} - 1 + x^{2} + 6 x + 18

= - \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} + 1 + \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} - 1 + x^{2} + 6 x + 18

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} + 1 + \frac{3}{2} ln \frac{x ^{2} + 6 x + 18}{3} - 1 + x^{2} + 6 x + 18 + C

\int \frac{x}{3 4 - x ^{2}} d x

\int coth (\frac{x}{4}) d x

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

\int (2 - 3 x^{2})^{2} d x

\int x^{2} - 3 x - 18 d x