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Beliebt Rechnen >

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

Lösung

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

Lösung

ln x + x^{2} + 1 + C

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

= ln x + x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= ln x + x^{2} + 1 + C

Graph

Beliebte Beispiele

steigung (-2/7 , 2/3),(5/7 ,-7/3)

s l o p e (- \frac{2}{7}, \frac{2}{3}), (\frac{5}{7}, - \frac{7}{3})

limit as x approaches 1 of (x-1)/(x^n-1)

x \to 1 lim (\frac{x - 1}{x ^{n} - 1})

(x+y)^2dx+(2xy+x^2-4)dy=0

(x + y)^{2} d x + (2 x y + x^{2} - 4) d y = 0

derivative of 5e^x+3

\frac{d}{d x} (5 e^{x} + 3)

derivative of (x^2+4e^{-2x})

\frac{d}{d x} ((x^{2} + 4) e^{- 2 x})