integral of (x^2-16)/(x(x+2)^3)

Lösung

\int \frac{x ^{2} - 16}{x ( x + 2 ) ^{3}} d x

- \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} - 2 ln (\frac{1}{2} ∣ x ∣) + 2 ln \frac{x}{2} + 1 + C

Schritte zur Lösung

\int \frac{x ^{2} - 16}{x ( x + 2 ) ^{3}} d x

Multipliziere aus

\frac{x ^{2} - 16}{x ( x + 2 ) ^{3}} : \frac{x}{( x + 2 ) ^{3}} - \frac{16}{x ( x + 2 ) ^{3}}

= \int \frac{x}{( x + 2 ) ^{3}} - \frac{16}{x ( x + 2 ) ^{3}} d x

Wende die Summenregel an:

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

= \int \frac{x}{( x + 2 ) ^{3}} d x - \int \frac{16}{x ( x + 2 ) ^{3}} d x

\int \frac{x}{( x + 2 ) ^{3}} d x = - \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}

\int \frac{16}{x ( x + 2 ) ^{3}} d x = 4 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

= - \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} - 4 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

Vereinfache

- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} - 4 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1) : - \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} - 2 ln (\frac{1}{2} ∣ x ∣) + 2 ln \frac{x}{2} + 1

= - \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} - 2 ln (\frac{1}{2} ∣ x ∣) + 2 ln \frac{x}{2} + 1

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} - 2 ln (\frac{1}{2} ∣ x ∣) + 2 ln \frac{x}{2} + 1 + C

\frac{\partial}{\partial x} (- x^{2} + y^{2})

t^{2} y^{^{''}} - 2 y = 3 t^{2} - 1

x \to 0 lim (\frac{( 1 )}{x})

y^{^{'}} = (1 - y) (2 - y)

(ln (e))^{^{'}}