integral of x/(sqrt(2-3x))

Lösung

\int \frac{x}{2 - 3 x} d x

\frac{2}{9} (\frac{1}{3} (2 - 3 x)^{\frac{3}{2}} - 2 2 - 3 x) + C

Schritte zur Lösung

\int \frac{x}{2 - 3 x} d x

Wende U-Substitution an

= \int \frac{2 ( u ^{2} - 2 )}{9} d u

Entferne die Konstante:

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

= \frac{2}{9} \cdot \int u^{2} - 2 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{2}{9} (\int u^{2} d u - \int 2 d u)

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

\int 2 d u = 2 u

= \frac{2}{9} (\frac{u ^{3}}{3} - 2 u)

Setze in

u = 2 - 3 x

ein

= \frac{2}{9} (\frac{( 2 - 3 x ) ^{3}}{3} - 2 2 - 3 x)

Vereinfache

\frac{2}{9} (\frac{( 2 - 3 x ) ^{3}}{3} - 2 2 - 3 x) : \frac{2}{9} (\frac{1}{3} (2 - 3 x)^{\frac{3}{2}} - 2 2 - 3 x)

= \frac{2}{9} (\frac{1}{3} (2 - 3 x)^{\frac{3}{2}} - 2 2 - 3 x)

Füge eine Konstante zur Lösung hinzu

= \frac{2}{9} (\frac{1}{3} (2 - 3 x)^{\frac{3}{2}} - 2 2 - 3 x) + C

\frac{d y}{d x} = (x - 10)^{2}

d er i v a t i v e e^{x} + 1

t an g e n t f (x) = x^{2} - 1, (5, 24)

\int_{0}^{\frac{π}{2}} (cos^{2} (x)) d x

t an g e n t 2 x^{2} + 8 x + 9