de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (2x^3+x^2+4)/((x^2+4)^2)

Lösung

\int \frac{2 x ^{3} + x ^{2} + 4}{( x ^{2} + 4 ) ^{2}} d x

Lösung

ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{4}{x ^{2} + 4} + C

Schritte zur Lösung

\int \frac{2 x ^{3} + x ^{2} + 4}{( x ^{2} + 4 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{2 x ^{3} + x ^{2} + 4}{( x ^{2} + 4 ) ^{2}} : \frac{2 x + 1}{x ^{2} + 4} - \frac{8 x}{( x ^{2} + 4 ) ^{2}}

= \int \frac{2 x + 1}{x ^{2} + 4} - \frac{8 x}{( x ^{2} + 4 ) ^{2}} 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{2 x + 1}{x ^{2} + 4} d x - \int \frac{8 x}{( x ^{2} + 4 ) ^{2}} d x

\int \frac{2 x + 1}{x ^{2} + 4} d x = ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2})

\int \frac{8 x}{( x ^{2} + 4 ) ^{2}} d x = - \frac{4}{x ^{2} + 4}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{4}{x ^{2} + 4} + C

Graph

Beliebte Beispiele

(\partial)/(\partial x)(xe^y-ln(x))

\frac{\partial}{\partial x} (x e^{y} - ln (x))

derivative of (4x^2-3/(x-1))

\frac{d}{d x} (\frac{4 x ^{2} - 3}{x - 1})

d/(dt)((sin(t)+cos(t))^2)

\frac{d}{d t} ((sin (t) + cos (t))^{2})

steigung y=5x^2+6/x

s l o p e y = 5 x^{2} + \frac{6}{x}

derivative y=(8t)/(8+sqrt(t))

d er i v a t i v e y = \frac{8 t}{8 + t}