integral of 4/((1+2x^2)^3)

Lösung

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

\frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= 4 \cdot \int \frac{1}{( 1 + 2 x ^{2} ) ^{3}} d x

Wende die partielle Integration an

= 4 (\frac{x}{( 1 + 2 x ^{2} ) ^{3}} - \int - \frac{12 x ^{2}}{( 1 + 2 x ^{2} ) ^{4}} d x)

\int - \frac{12 x ^{2}}{( 1 + 2 x ^{2} ) ^{4}} d x = - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

= 4 \frac{x}{( 1 + 2 x ^{2} ) ^{3}} - - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

Vereinfache

4 \frac{x}{( 1 + 2 x ^{2} ) ^{3}} - - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} : \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

= \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

Füge eine Konstante zur Lösung hinzu

= \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} + C

\int \frac{4 x ^{2} + x + 17}{( x + 1 ) ( x ^{2} + 4 )} d x

\int (x - 1) (2 - x) d x

\int \frac{( x ^{2} - 2 )}{x ^{3} - 6 x + 2} d x

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

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