de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of x/(x^2+2x+10)

Lösung

\int \frac{x}{x ^{2} + 2 x + 10} d x

Lösung

\frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 2 x + 10} d x

Vervollständige das Quadrat

x^{2} + 2 x + 10 : (x + 1)^{2} + 9

= \int \frac{x}{( x + 1 ) ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{u - 1}{u ^{2} + 9} d u

Multipliziere aus

\frac{u - 1}{u ^{2} + 9} : \frac{u}{u ^{2} + 9} - \frac{1}{u ^{2} + 9}

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{u}{u ^{2} + 9} d u - \int \frac{1}{u ^{2} + 9} d u

\int \frac{u}{u ^{2} + 9} d u = \frac{1}{2} ln u^{2} + 9

\int \frac{1}{u ^{2} + 9} d u = \frac{1}{3} arctan (\frac{u}{3})

= \frac{1}{2} ln u^{2} + 9 - \frac{1}{3} arctan (\frac{u}{3})

Setze in

u = x + 1

ein

= \frac{1}{2} ln (x + 1)^{2} + 9 - \frac{1}{3} arctan (\frac{x + 1}{3})

Vereinfache

\frac{1}{2} ln (x + 1)^{2} + 9 - \frac{1}{3} arctan (\frac{x + 1}{3}) : \frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

= \frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Graph

Beliebte Beispiele

integral of (x^4)/((1+x^2))

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

integral of sec(x)tan(x)cos(sec(x))

\int sec (x) tan (x) cos (sec (x)) d x

integral of 9xsqrt(6x-3)

\int 9 x 6 x - 3 d x

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

\int \frac{x ^{2}}{3 + x ^{3}} d x

integral of 8xsqrt(x^2+1)

\int 8 x x^{2} + 1 d x