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

Lösung

integral

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

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

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} + 2 x : (x + 1)^{2} - 1

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

Wende U-Substitution an

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

Wende die partielle Integration an

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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

in t e g r a l \frac{v}{2}

in t e g r a l \frac{( x - 1 )}{( x ^{2} + 2 x + 3 )}

in t e g r a l \frac{1}{x ( x ^{2} - 1 )}

in t e g r a l \frac{x ^{- 2} ( e ^{x} ) ^{3}}{3}