integral (x+(x^2+1)^{1/2})^{-1}

Lösung

integral

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

\frac{1}{2} ln x + (x^{2} + 1)^{\frac{1}{2}} - \frac{1}{2 ( x + ( x ^{2} + 1 ) ^{\frac{1}{2}} ) ^{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u ^{2} + 1}{2 u ^{3}} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u ^{2} + 1}{u ^{3}} d u

Multipliziere aus

\frac{u ^{2} + 1}{u ^{3}} : \frac{1}{u} + \frac{1}{u ^{3}}

= \frac{1}{2} \cdot \int \frac{1}{u} + \frac{1}{u ^{3}} d u

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

= \frac{1}{2} (ln ∣ u ∣ - \frac{1}{2 u ^{2}})

Setze in

u = x + (x^{2} + 1)^{\frac{1}{2}}

ein

= \frac{1}{2} ln x + (x^{2} + 1)^{\frac{1}{2}} - \frac{1}{2 ( x + ( x ^{2} + 1 ) ^{\frac{1}{2}} ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x + (x^{2} + 1)^{\frac{1}{2}} - \frac{1}{2 ( x + ( x ^{2} + 1 ) ^{\frac{1}{2}} ) ^{2}} + C

in t e g r a l (1 - x)^{5}

in t e g r a l \frac{1 + 1}{x ^{4}}

in t e g r a l \frac{e ^{x} + 1}{e ^{x}}

in t e g r a l e^{7} y

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