integral ((3x^2+5))/((x^2+1)^2)

Lösung

integral

\frac{( 3 x ^{2} + 5 )}{( x ^{2} + 1 ) ^{2}}

- \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 5}{( x ^{2} + 1 ) ^{2}} d x

Multipliziere aus

\frac{3 x ^{2} + 5}{( x ^{2} + 1 ) ^{2}} : \frac{3 x ^{2}}{( x ^{2} + 1 ) ^{2}} + \frac{5}{( x ^{2} + 1 ) ^{2}}

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{3 x ^{2}}{( x ^{2} + 1 ) ^{2}} d x + \int \frac{5}{( x ^{2} + 1 ) ^{2}} d x

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

\int \frac{5}{( x ^{2} + 1 ) ^{2}} d x = \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

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

Vereinfache

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

= - \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1}

Füge eine Konstante zur Lösung hinzu

= - \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + C

in t e g r a l 2 - (\frac{5}{x ^{2}})

in t e g r a l 1 - x^{2}

in t e g r a l ∣ x ∣^{4}

in t e g r a l \frac{7}{x}

in t e g r a l \frac{2}{( ( x - 2 ) x + 2 )}