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Beliebt Rechnen >

integral of x^2sqrt(2x+5)

Lösung

\int x^{2} 2 x + 5 d x

Lösung

\frac{1}{21} (3 x^{2} - 6 x + 10) (2 x + 5)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{2} 2 x + 5 d x

Wende U-Substitution an

= \int \frac{u ( u - 5 ) ^{2}}{8} d u

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int u (u - 5)^{2} d u

Multipliziere aus

u (u - 5)^{2} : u^{\frac{5}{2}} - 10 u^{\frac{3}{2}} + 25 u

= \frac{1}{8} \cdot \int u^{\frac{5}{2}} - 10 u^{\frac{3}{2}} + 25 u 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

= \frac{1}{8} (\int u^{\frac{5}{2}} d u - \int 10 u^{\frac{3}{2}} d u + \int 25 u d u)

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 10 u^{\frac{3}{2}} d u = 4 u^{\frac{5}{2}}

\int 25 u d u = \frac{50}{3} u^{\frac{3}{2}}

= \frac{1}{8} (\frac{2}{7} u^{\frac{7}{2}} - 4 u^{\frac{5}{2}} + \frac{50}{3} u^{\frac{3}{2}})

Setze in

u = 2 x + 5

ein

= \frac{1}{8} (\frac{2}{7} (2 x + 5)^{\frac{7}{2}} - 4 (2 x + 5)^{\frac{5}{2}} + \frac{50}{3} (2 x + 5)^{\frac{3}{2}})

Vereinfache

\frac{1}{8} (\frac{2}{7} (2 x + 5)^{\frac{7}{2}} - 4 (2 x + 5)^{\frac{5}{2}} + \frac{50}{3} (2 x + 5)^{\frac{3}{2}}) : \frac{1}{21} (3 x^{2} - 6 x + 10) (2 x + 5)^{\frac{3}{2}}

= \frac{1}{21} (3 x^{2} - 6 x + 10) (2 x + 5)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{21} (3 x^{2} - 6 x + 10) (2 x + 5)^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

tangent y= 3/(sqrt(x)),(4, 3/2)

t an g e n t y = \frac{3}{x}, (4, \frac{3}{2})

laplacetransform t^2cos(t)

l a pl a ce t r an s f or m t^{2} cos (t)

y^{''}+4y^'+4y=t+e^{2t}

y^{^{''}} + 4 y^{^{'}} + 4 y = t + e^{2 t}

tangent f(x)=9x^2

t an g e n t f (x) = 9 x^{2}

integral of-2x^3+12x^2-20x+8.5

\int - 2 x^{3} + 12 x^{2} - 20 x + 8.5 d x