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)/(sqrt((x-1)^2+4))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

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

Setze in

u = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

y^{'''}-y^{''}+y^'+3y=0

y^{^{'''}} - y^{^{''}} + y^{^{'}} + 3 y = 0

(dy)/(dt)=3y+y^4,y(0)=1

\frac{d y}{d t} = 3 y + y^{4}, y (0) = 1

derivative of-20x^3+36x^2-12x-4

\frac{d}{d x} (- 20 x^{3} + 36 x^{2} - 12 x - 4)

(d^2)/(dx^2)(9z^2e^z)

\frac{d ^{2}}{d x ^{2}} (9 z^{2} e^{z})

integral from 0 to 8 of 1/(64+x^2)

\int_{0}^{8} \frac{1}{64 + x ^{2}} d x