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Beliebt Rechnen >

integral of 6/(xsqrt(x^2+1))

Lösung

\int \frac{6}{x x ^{2} + 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{6}{x x ^{2} + 1} d x

Entferne die Konstante:

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

= 6 \cdot \int \frac{1}{x x ^{2} + 1} d x

Wende U-Substitution an

= 6 \cdot \int \frac{1}{u ^{2} - 1} d u

Faktorisiere

u^{2} - 1 : - (- u^{2} + 1)

= 6 \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 6 (- \int \frac{1}{- u ^{2} + 1} d u)

Nutze das gemeinsame Integral :

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

= 6 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Setze in

u = x^{2} + 1

ein

= 6 (- (\frac{ln x ^{2} + 1 + 1}{2} - \frac{ln x ^{2} + 1 - 1}{2}))

Vereinfache

6 (- (\frac{ln x ^{2} + 1 + 1}{2} - \frac{ln x ^{2} + 1 - 1}{2})) : - 6 (\frac{1}{2} ln x^{2} + 1 + 1 - \frac{1}{2} ln x^{2} + 1 - 1)

= - 6 (\frac{1}{2} ln x^{2} + 1 + 1 - \frac{1}{2} ln x^{2} + 1 - 1)

Füge eine Konstante zur Lösung hinzu

= - 6 (\frac{1}{2} ln x^{2} + 1 + 1 - \frac{1}{2} ln x^{2} + 1 - 1) + C

Graph

Beliebte Beispiele

integral of (x+1)e^{9x^2+18x}

\int (x + 1) e^{9 x^{2} + 18 x} d x

(d^2y)/(dx^2)-4(dy)/(dx)+13y=8sin(3x),y(0)=1,y^'(0)=2

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 13 y = 8 sin (3 x), y (0) = 1, y^{^{'}} (0) = 2

limit as x approaches 1 of (9x^2)+30x+25

x \to 1 lim ((9 x^{2}) + 30 x + 25)

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\int x^{25} sin (x^{26}) d x