integral of rctan(sqrt(x))

Lösung

\int d arctan (x) d x

d x arctan (x) + d arctan (x) - d x + C

Schritte zur Lösung

\int d arctan (x) d x

Entferne die Konstante:

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

= d \cdot \int arctan (x) d x

Wende U-Substitution an

= d \cdot \int arctan (u) \cdot 2 u d u

Entferne die Konstante:

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

= d \cdot 2 \cdot \int arctan (u) u d u

Wende die partielle Integration an

= d \cdot 2 (\frac{1}{2} u^{2} arctan (u) - \int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u)

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

= d \cdot 2 (\frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u))

Setze in

u = x

ein

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

Vereinfache

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

= d x arctan (x) + d arctan (x) - d x

Füge eine Konstante zur Lösung hinzu

= d x arctan (x) + d arctan (x) - d x + C

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

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

\int \frac{1}{5 x - x ^{2}} d x

\int \frac{x ^{3} + x ^{2}}{x ^{2} + x - 2} d x

\int (2 x^{5} - 20 x)^{3} (x^{4} - 2) d x