integral of (arctan(x))/((1+x^2)^{3/2)}

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int 12 x^{2} (x^{3} + 4)^{3} d x

\frac{d}{d x} (x 2)

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

a re a x, x^{1}, 0, 1

n \to \infty lim (\frac{n}{4})