integral of (3x^2+4)sqrt(x-1)

Lösung

\int (3 x^{2} + 4) x - 1 d x

\frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (3 x^{2} + 4) x - 1 d x

Wende die partielle Integration an

= 2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - \int 4 x (x - 1)^{\frac{3}{2}} d x

\int 4 x (x - 1)^{\frac{3}{2}} d x = 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}})

= 2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}})

Vereinfache

2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}}) : \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}}

= \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}} + C

\int_{4}^{7} (x - 5)^{2} d x

y^{^{'}} + y = e^{x}, y (0) = 6

\frac{\partial}{\partial x} (\frac{x}{x ^{5} - y ^{5}})

\int sin (x) csc (x) d x

f (x) = e^{2 x^{2}}