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

Lösung

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

(1 + y^{2})^{\frac{1}{2}} + C

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

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

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

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Setze in

u = 1 + y^{2}

ein

= \frac{1}{2} \cdot 2 (1 + y^{2})^{\frac{1}{2}}

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= (1 + y^{2})^{\frac{1}{2}} + C

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