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)/((121+x^2)^2)

Lösung

\int \frac{x ^{2}}{( 121 + x ^{2} ) ^{2}} d x

Lösung

\frac{1}{22} arctan (\frac{x}{11}) - \frac{x}{2 ( 121 + x ^{2} )} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( 121 + x ^{2} ) ^{2}} d x

Wende integrale Substitution an

= \int \frac{u ^{2}}{11 ( 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}{11} \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}{11} \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}{11} (\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}{11} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Setze in

u = \frac{x}{11}

ein

= \frac{1}{11} (arctan (\frac{x}{11}) - \frac{1}{2} (arctan (\frac{x}{11}) + \frac{\frac{x}{11}}{1 + ( \frac{x}{11} ) ^{2}}))

Vereinfache

\frac{1}{11} (arctan (\frac{x}{11}) - \frac{1}{2} (arctan (\frac{x}{11}) + \frac{\frac{x}{11}}{1 + ( \frac{x}{11} ) ^{2}})) : \frac{1}{22} arctan (\frac{x}{11}) - \frac{x}{2 ( 121 + x ^{2} )}

= \frac{1}{22} arctan (\frac{x}{11}) - \frac{x}{2 ( 121 + x ^{2} )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{22} arctan (\frac{x}{11}) - \frac{x}{2 ( 121 + x ^{2} )} + C

Graph

Beliebte Beispiele

integral of (y^3e^{y^2})/((y^2+1)^2)

\int \frac{y ^{3} e ^{y^{2}}}{( y ^{2} + 1 ) ^{2}} d y

fläche f(x)=x^2-2x,(0,2)

a re a f (x) = x^{2} - 2 x, (0, 2)

derivative of x-3\sqrt[3]{x}

\frac{d}{d x} (x - 3 3 x)

y^{''}-8y^'+15y=4e^t

y^{^{''}} - 8 y^{^{'}} + 15 y = 4 e^{t}

(\partial)/(\partial x)(2x+3e^y+xy)

\frac{\partial}{\partial x} (2 x + 3 e^{y} + x y)