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Beliebt Rechnen >

integral of (3x^2)/(sqrt(x^2+4x+5))

Lösung

\int \frac{3 x ^{2}}{x ^{2} + 4 x + 5} d x

Lösung

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

Schritte zur Lösung

\int \frac{3 x ^{2}}{x ^{2} + 4 x + 5} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{x ^{2}}{x ^{2} + 4 x + 5} d x

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

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

Setze in

u = x + 2

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(x^2)*(1/t-1)^{1/2}

\int \frac{1}{x ^{2}} \cdot (\frac{1}{t} - 1)^{\frac{1}{2}}

integral of 1/((sin(a)y)(cos(a)y))

\int \frac{1}{( sin ( a ) y ) ( cos ( a ) y )} d y

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

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

integral of (x^2-1)/(x^3-5x^2+x-5)

\int \frac{x ^{2} - 1}{x ^{3} - 5 x ^{2} + x - 5} d x

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

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