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

Lösung

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

\frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}} + C

Schritte zur Lösung

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

Wende integrale Reduktion an

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

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

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

Vereinfache

\frac{3}{4} \cdot \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{2 \cdot 2 ( x ^{2} + 1 ) ^{2}} : \frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}}

= \frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}} + C

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