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Beliebt Rechnen >

integral of xsqrt(a+bx)

Lösung

\int x a + b x d x

Lösung

\frac{2 ( 5 b x ( a + b x ) ^{\frac{3}{2}} - 2 ( a + b x ) ^{\frac{5}{2}} )}{15 b ^{2}} + C

Schritte zur Lösung

\int x a + b x d x

Wende U-Substitution an

= \int \frac{2 u ^{2} ( u ^{2} - a )}{b ^{2}} d u

Entferne die Konstante:

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

= \frac{2}{b ^{2}} \cdot \int u^{2} (u^{2} - a) d u

Wende die partielle Integration an

= \frac{2}{b ^{2}} (\frac{1}{3} u^{3} (u^{2} - a) - \int \frac{2 u ^{4}}{3} d u)

\int \frac{2 u ^{4}}{3} d u = \frac{2 u ^{5}}{15}

= \frac{2}{b ^{2}} (\frac{1}{3} u^{3} (u^{2} - a) - \frac{2 u ^{5}}{15})

Setze in

u = a + b x

ein

= \frac{2}{b ^{2}} (\frac{1}{3} (a + b x)^{3} ((a + b x)^{2} - a) - \frac{2 ( a + b x ) ^{5}}{15})

Vereinfache

\frac{2}{b ^{2}} (\frac{1}{3} (a + b x)^{3} ((a + b x)^{2} - a) - \frac{2 ( a + b x ) ^{5}}{15}) : \frac{2 ( 5 b x ( a + b x ) ^{\frac{3}{2}} - 2 ( a + b x ) ^{\frac{5}{2}} )}{15 b ^{2}}

= \frac{2 ( 5 b x ( a + b x ) ^{\frac{3}{2}} - 2 ( a + b x ) ^{\frac{5}{2}} )}{15 b ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2 ( 5 b x ( a + b x ) ^{\frac{3}{2}} - 2 ( a + b x ) ^{\frac{5}{2}} )}{15 b ^{2}} + C

Graph

Beliebte Beispiele

integral of (2x-4x^3+5)

\int (2 x - 4 x^{3} + 5) d x

integral of sec^2(x/2)tan^2(x/2)

\int sec^{2} (\frac{x}{2}) tan^{2} (\frac{x}{2}) d x

integral of cos(5y)

\int cos (5 y) d y

integral of (e^x+6)/(e^x+6x)

\int \frac{e ^{x} + 6}{e ^{x} + 6 x} d x

integral of θtan(9)θ^2

\int θ tan (9) θ^{2} d θ