de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (3x+2)/(x^2+2x+5)

Lösung

\int \frac{3 x + 2}{x ^{2} + 2 x + 5} d x

Lösung

\frac{3}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1)) + C

Schritte zur Lösung

\int \frac{3 x + 2}{x ^{2} + 2 x + 5} d x

Vervollständige das Quadrat

x^{2} + 2 x + 5 : (x + 1)^{2} + 4

= \int \frac{3 x + 2}{( x + 1 ) ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{3 u - 1}{u ^{2} + 4} d u

Multipliziere aus

\frac{3 u - 1}{u ^{2} + 4} : \frac{3 u}{u ^{2} + 4} - \frac{1}{u ^{2} + 4}

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

\int \frac{3 u}{u ^{2} + 4} d u = \frac{3}{2} ln u^{2} + 4

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

= \frac{3}{2} ln u^{2} + 4 - \frac{1}{2} arctan (\frac{u}{2})

Setze in

u = x + 1

ein

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

Vereinfache

\frac{3}{2} ln (x + 1)^{2} + 4 - \frac{1}{2} arctan (\frac{x + 1}{2}) : \frac{3}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))

= \frac{3}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1)) + C

Graph

Beliebte Beispiele

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

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

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

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

integral of (4y-x^2)^2

\int (4 y - x^{2})^{2} d x

integral of (x^2)/(sqrt(25-16x^2))

\int \frac{x ^{2}}{25 - 16 x ^{2}} d x

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

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