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Beliebt Rechnen >

integral of xsqrt(ax+b)

Lösung

\int x a x + b d x

Lösung

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

Schritte zur Lösung

\int x a x + b d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Setze in

u = a x + b

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(sqrt(2θ-θ^2))

\int \frac{1}{2 θ - θ ^{2}} d θ

integral of 1/(sqrt(1-9t^2))

\int \frac{1}{1 - 9 t ^{2}} d t

integral of arccosh(2x)

\int arccosh (2 x) d x

integral of (x^2+44x-15)/(2x^3+6x)

\int \frac{x ^{2} + 44 x - 15}{2 x ^{3} + 6 x} d x

integral of (cos(pi/(x^{25)}))/(x^{26)}

\int \frac{cos ( \frac{π}{x ^{25}} )}{x ^{26}} d x