integral of (5x^2)/((81+x^2)^2)

Lösung

\int \frac{5 x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

\frac{5 ( x ^{2} arctan ( \frac{x}{9} ) - 9 x + 81 arctan ( \frac{x}{9} ) )}{18 ( 81 + x ^{2} )} + C

Schritte zur Lösung

\int \frac{5 x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

Wende integrale Substitution an

= 5 \cdot \int \frac{u ^{2}}{9 ( u ^{2} + 1 ) ^{2}} d u

Entferne die Konstante:

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

= 5 \cdot \frac{1}{9} \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}}

= 5 \cdot \frac{1}{9} \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

= 5 \cdot \frac{1}{9} (\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}})

= 5 \cdot \frac{1}{9} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Setze in

u = \frac{x}{9}

ein

= 5 \cdot \frac{1}{9} (arctan (\frac{x}{9}) - \frac{1}{2} (arctan (\frac{x}{9}) + \frac{\frac{x}{9}}{1 + ( \frac{x}{9} ) ^{2}}))

Vereinfache

5 \cdot \frac{1}{9} (arctan (\frac{x}{9}) - \frac{1}{2} (arctan (\frac{x}{9}) + \frac{\frac{x}{9}}{1 + ( \frac{x}{9} ) ^{2}})) : \frac{5 ( x ^{2} arctan ( \frac{x}{9} ) - 9 x + 81 arctan ( \frac{x}{9} ) )}{18 ( 81 + x ^{2} )}

= \frac{5 ( x ^{2} arctan ( \frac{x}{9} ) - 9 x + 81 arctan ( \frac{x}{9} ) )}{18 ( 81 + x ^{2} )}

Füge eine Konstante zur Lösung hinzu

= \frac{5 ( x ^{2} arctan ( \frac{x}{9} ) - 9 x + 81 arctan ( \frac{x}{9} ) )}{18 ( 81 + x ^{2} )} + C

\int \frac{2 x ^{2} - 2 x + 12}{x ^{3} + 3 x} d x

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