integral ((r^2))/((1+r^2)d)

Lösung

integral

\frac{( r ^{2} )}{( 1 + r ^{2} ) d}

\frac{1}{d} (- arctan (r) + r) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

= \frac{1}{d} \cdot \int \frac{r ^{2}}{1 + r ^{2}} d r

\frac{r ^{2}}{1 + r ^{2}} = - \frac{1}{1 + r ^{2}} + 1

= \frac{1}{d} \cdot \int - \frac{1}{1 + r ^{2}} + 1 d r

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d r = r

= \frac{1}{d} (- arctan (r) + r)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{d} (- arctan (r) + r) + C

in t e g r a l sin^{3} (2 x) cos (2 x)

in t e g r a l (1 + 8 x)^{\frac{3}{2}}

in t e g r a l \frac{( x ^{2} - x + 1 )}{x ^{3}}

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

in t e g r a l \frac{1}{( 5 + 2 x + x ^{2} ) ^{\frac{3}{2}}}