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

Lösung

\int \frac{e ^{a r c t a n (x)}}{( 1 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{e ^{a r c t a n (x)} x}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{e ^{a r c t a n (x)}}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + C

Schritte zur Lösung

\int \frac{e ^{a r c t a n (x)}}{( 1 + x ^{2} ) ^{\frac{3}{2}}} d x

Wende U-Substitution an

= \int e^{u} cos (u) d u

Wende die partielle Integration an

= e^{u} sin (u) - \int e^{u} sin (u) d u

Wende die partielle Integration an

= e^{u} sin (u) - (- e^{u} cos (u) - \int - e^{u} cos (u) d u)

Entferne die Konstante:

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

= e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u))

Deshalb

\int e^{u} cos (u) d u = e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u))

Isoliere

\int e^{u} cos (u) d u

= \frac{e ^{u} sin ( u )}{2} + \frac{e ^{u} cos ( u )}{2}

Setze in

u = arctan (x)

ein

= \frac{e ^{a r c t a n (x)} sin ( arctan ( x ) )}{2} + \frac{e ^{a r c t a n (x)} cos ( arctan ( x ) )}{2}

Vereinfache

\frac{e ^{a r c t a n (x)} sin ( arctan ( x ) )}{2} + \frac{e ^{a r c t a n (x)} cos ( arctan ( x ) )}{2} : \frac{e ^{a r c t a n (x)} x}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{e ^{a r c t a n (x)}}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{e ^{a r c t a n (x)} x}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{e ^{a r c t a n (x)}}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}}

Füge eine Konstante zur Lösung hinzu

= \frac{e ^{a r c t a n (x)} x}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{e ^{a r c t a n (x)}}{2 ( 1 + x ^{2} ) ^{\frac{1}{2}}} + C

\int \frac{x sin ( x )}{cos ^{3} ( x )} d x

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

\int x e^{x} + x d x

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

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