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Beliebt Rechnen >

integral of (x/(sqrt(x+1)))

Lösung

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

Lösung

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

Schritte zur Lösung

\int \frac{x}{x + 1} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 2 \cdot \int 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

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

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= 2 (\frac{u ^{3}}{3} - u)

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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(\partial)/(\partial x)(y-1)

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x \to - \infty lim (5)

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\int (e^{- x})^{2} d x

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inverselaplace (s^2+2s+3)/((s+1)^3)

in v erse l a pl a ce \frac{s ^{2} + 2 s + 3}{( s + 1 ) ^{3}}