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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = 2 x

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

3y^{''}+13y^'+4y=0,y(0)=-2,y^'(0)=-2

3 y^{^{''}} + 13 y^{^{'}} + 4 y = 0, y (0) = - 2, y^{^{'}} (0) = - 2

integral of (5x^3-3x^2+7)

\int (5 x^{3} - 3 x^{2} + 7) d x

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\frac{d}{d x} (x^{4} - 8 x^{2} + 16)

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\frac{d}{d y} (\frac{1}{y ^{3}})

(\partial)/(\partial x)(2x^2+5y^2-xy)

\frac{\partial}{\partial x} (2 x^{2} + 5 y^{2} - x y)