integral of x/(sqrt(x^2+x+2))

Lösung

\int \frac{x}{x ^{2} + x + 2} d x

x^{2} + x + 2 - \frac{1}{2} ln \frac{2}{7} x + \frac{1}{7} + \frac{4 x ^{2} + 4 x + 8}{7} + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + x + 2} d x

Vervollständige das Quadrat

x^{2} + x + 2 : (x + \frac{1}{2})^{2} + \frac{7}{4}

= \int \frac{x}{( x + \frac{1}{2} ) ^{2} + \frac{7}{4}} d x

Wende U-Substitution an

= \int \frac{2 u - 1}{4 u ^{2} + 7} d u

Multipliziere aus

\frac{2 u - 1}{4 u ^{2} + 7} : \frac{2 u}{4 u ^{2} + 7} - \frac{1}{4 u ^{2} + 7}

= \int \frac{2 u}{4 u ^{2} + 7} - \frac{1}{4 u ^{2} + 7} 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

= \int \frac{2 u}{4 u ^{2} + 7} d u - \int \frac{1}{4 u ^{2} + 7} d u

\int \frac{2 u}{4 u ^{2} + 7} d u = \frac{1}{2} 4 u^{2} + 7

\int \frac{1}{4 u ^{2} + 7} d u = \frac{1}{2} ln \frac{2}{7} u + \frac{1}{7} (7 + 4 u^{2})

= \frac{1}{2} 4 u^{2} + 7 - \frac{1}{2} ln \frac{2}{7} u + \frac{1}{7} (7 + 4 u^{2})

Setze in

u = x + \frac{1}{2}

ein

= \frac{1}{2} 4 (x + \frac{1}{2})^{2} + 7 - \frac{1}{2} ln \frac{2}{7} (x + \frac{1}{2}) + \frac{1}{7} (7 + 4 (x + \frac{1}{2})^{2})

Vereinfache

\frac{1}{2} 4 (x + \frac{1}{2})^{2} + 7 - \frac{1}{2} ln \frac{2}{7} (x + \frac{1}{2}) + \frac{1}{7} (7 + 4 (x + \frac{1}{2})^{2}) : x^{2} + x + 2 - \frac{1}{2} ln \frac{2}{7} x + \frac{1}{7} + \frac{4 x ^{2} + 4 x + 8}{7}

= x^{2} + x + 2 - \frac{1}{2} ln \frac{2}{7} x + \frac{1}{7} + \frac{4 x ^{2} + 4 x + 8}{7}

Füge eine Konstante zur Lösung hinzu

= x^{2} + x + 2 - \frac{1}{2} ln \frac{2}{7} x + \frac{1}{7} + \frac{4 x ^{2} + 4 x + 8}{7} + C

\int 7 t d t

\int sin (7 π t) d t

\frac{d}{d x} (csc (x) + cot (x))

l a pl a ce t r an s f or m e^{- t} sin (3 t)

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