integral of xsqrt(17x-14)

Lösung

\int x 17 x - 14 d x

\frac{1}{289} (\frac{2}{5} (17 x - 14)^{\frac{5}{2}} + \frac{28}{3} (17 x - 14)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x 17 x - 14 d x

Wende U-Substitution an

= \int \frac{u ( u + 14 )}{289} d u

Entferne die Konstante:

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

= \frac{1}{289} \cdot \int u (u + 14) d u

Multipliziere aus

u (u + 14) : u^{\frac{3}{2}} + 14 u

= \frac{1}{289} \cdot \int u^{\frac{3}{2}} + 14 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}{289} (\int u^{\frac{3}{2}} d u + \int 14 u d u)

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

\int 14 u d u = \frac{28}{3} u^{\frac{3}{2}}

= \frac{1}{289} (\frac{2}{5} u^{\frac{5}{2}} + \frac{28}{3} u^{\frac{3}{2}})

Setze in

u = 17 x - 14

ein

= \frac{1}{289} (\frac{2}{5} (17 x - 14)^{\frac{5}{2}} + \frac{28}{3} (17 x - 14)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{289} (\frac{2}{5} (17 x - 14)^{\frac{5}{2}} + \frac{28}{3} (17 x - 14)^{\frac{3}{2}}) + C

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