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Beliebt Rechnen >

integral of ((x/3+1/3))/((x^2+x+1)^2)

Lösung

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

Lösung

\frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))) + C

Schritte zur Lösung

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

Vereinfache

\frac{\frac{x}{3} + \frac{1}{3}}{( x ^{2} + x + 1 ) ^{2}} : \frac{x + 1}{3 ( x ^{2} + x + 1 ) ^{2}}

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

\int \frac{x}{( x ^{2} + x + 1 ) ^{2}} d x = 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))))

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

= \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))) + C

Graph

Beliebte Beispiele

integral of x^3sqrt(1+(3x^2)^2)

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

integral of (xln(x^2+3))/(x^2+3)

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

integral of sqrt(1^2-x^2)

\int 1^{2} - x^{2} d x

integral of 1/(x^2sqrt(x+1))

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

integral of 1/(sqrt(x)+9)

\int \frac{1}{x + 9} d x