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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{\partial}{\partial y} (\frac{1}{2} (x - y)^{2})

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

in v erse l a pl a ce \frac{1}{2 s ^{2} + s}

\int \frac{1}{x ^{1}} d x

\int \frac{sin ( 1 + x )}{x} d x