integral of 1/(y^2(y^2+1))

Lösung

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

- \frac{1}{y} - arctan (y) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{y ^{2} ( y ^{2} + 1 )} : \frac{1}{y ^{2}} - \frac{1}{y ^{2} + 1}

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

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{1}{y ^{2}} d y - \int \frac{1}{y ^{2} + 1} d y

\int \frac{1}{y ^{2}} d y = - \frac{1}{y}

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

= - \frac{1}{y} - arctan (y)

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e e^{4 x^{2}}

x \to 4 - lim (\frac{2}{x - 4})

x \to - \infty lim (4^{x})

x \to - 4 lim (\frac{x ^{2} + 5 x + 4}{x ^{2} + 3 x - 4})

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