de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (x^2)/((225+x^2)^2)

Lösung

\int \frac{x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

Lösung

\frac{1}{30} arctan (\frac{x}{15}) - \frac{x}{2 ( 225 + x ^{2} )} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

Wende integrale Substitution an

= \int \frac{u ^{2}}{15 ( u ^{2} + 1 ) ^{2}} d u

Entferne die Konstante:

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

= \frac{1}{15} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

Ermittle den Partialbruch von

\frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} : \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}}

= \frac{1}{15} \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} 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

= \frac{1}{15} (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int \frac{1}{( u ^{2} + 1 ) ^{2}} d u = \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

= \frac{1}{15} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Setze in

u = \frac{x}{15}

ein

= \frac{1}{15} (arctan (\frac{x}{15}) - \frac{1}{2} (arctan (\frac{x}{15}) + \frac{\frac{x}{15}}{1 + ( \frac{x}{15} ) ^{2}}))

Vereinfache

\frac{1}{15} (arctan (\frac{x}{15}) - \frac{1}{2} (arctan (\frac{x}{15}) + \frac{\frac{x}{15}}{1 + ( \frac{x}{15} ) ^{2}})) : \frac{1}{30} arctan (\frac{x}{15}) - \frac{x}{2 ( 225 + x ^{2} )}

= \frac{1}{30} arctan (\frac{x}{15}) - \frac{x}{2 ( 225 + x ^{2} )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{30} arctan (\frac{x}{15}) - \frac{x}{2 ( 225 + x ^{2} )} + C

Graph

Beliebte Beispiele

integral of (1+x)/(x-2x+2x^2)

\int \frac{1 + x}{x - 2 x + 2 x ^{2}} d x

integral of 4/(4x^2-4x+5)

\int \frac{4}{4 x ^{2} - 4 x + 5} d x

limit as x approaches 0 of 2x+sqrt(x)

x \to 0 lim (2 x + x)

integral of e^{-2θ}cos(θ)

\int e^{- 2 θ} cos (θ) d θ

limit as x approaches 0 of xcos(2/x)

x \to 0 lim (x cos (\frac{2}{x}))