integral of 5x^4(2x^5-3)^8

Lösung

\int 5 x^{4} (2 x^{5} - 3)^{8} d x

\frac{256 x ^{45}}{9} - 384 x^{40} + 2304 x^{35} - 8064 x^{30} + 18144 x^{25} - 27216 x^{20} + 27216 x^{15} - 17496 x^{10} + 6561 x^{5} + C

Schritte zur Lösung

\int 5 x^{4} (2 x^{5} - 3)^{8} d x

Entferne die Konstante:

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

= 5 \cdot \int x^{4} (2 x^{5} - 3)^{8} d x

Multipliziere aus

x^{4} (2 x^{5} - 3)^{8} : 256 x^{44} - 3072 x^{39} + 16128 x^{34} - 48384 x^{29} + 90720 x^{24} - 108864 x^{19} + 81648 x^{14} - 34992 x^{9} + 6561 x^{4}

= 5 \cdot \int 256 x^{44} - 3072 x^{39} + 16128 x^{34} - 48384 x^{29} + 90720 x^{24} - 108864 x^{19} + 81648 x^{14} - 34992 x^{9} + 6561 x^{4} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (\int 256 x^{44} d x - \int 3072 x^{39} d x + \int 16128 x^{34} d x - \int 48384 x^{29} d x + \int 90720 x^{24} d x - \int 108864 x^{19} d x + \int 81648 x^{14} d x - \int 34992 x^{9} d x + \int 6561 x^{4} d x)

\int 256 x^{44} d x = \frac{256 x ^{45}}{45}

\int 3072 x^{39} d x = \frac{384 x ^{40}}{5}

\int 16128 x^{34} d x = \frac{2304 x ^{35}}{5}

\int 48384 x^{29} d x = \frac{8064 x ^{30}}{5}

\int 90720 x^{24} d x = \frac{18144 x ^{25}}{5}

\int 108864 x^{19} d x = \frac{27216 x ^{20}}{5}

\int 81648 x^{14} d x = \frac{27216 x ^{15}}{5}

\int 34992 x^{9} d x = \frac{17496 x ^{10}}{5}

\int 6561 x^{4} d x = \frac{6561 x ^{5}}{5}

= 5 (\frac{256 x ^{45}}{45} - \frac{384 x ^{40}}{5} + \frac{2304 x ^{35}}{5} - \frac{8064 x ^{30}}{5} + \frac{18144 x ^{25}}{5} - \frac{27216 x ^{20}}{5} + \frac{27216 x ^{15}}{5} - \frac{17496 x ^{10}}{5} + \frac{6561 x ^{5}}{5})

Vereinfache

5 (\frac{256 x ^{45}}{45} - \frac{384 x ^{40}}{5} + \frac{2304 x ^{35}}{5} - \frac{8064 x ^{30}}{5} + \frac{18144 x ^{25}}{5} - \frac{27216 x ^{20}}{5} + \frac{27216 x ^{15}}{5} - \frac{17496 x ^{10}}{5} + \frac{6561 x ^{5}}{5}) : \frac{256 x ^{45}}{9} - 384 x^{40} + 2304 x^{35} - 8064 x^{30} + 18144 x^{25} - 27216 x^{20} + 27216 x^{15} - 17496 x^{10} + 6561 x^{5}

= \frac{256 x ^{45}}{9} - 384 x^{40} + 2304 x^{35} - 8064 x^{30} + 18144 x^{25} - 27216 x^{20} + 27216 x^{15} - 17496 x^{10} + 6561 x^{5}

Füge eine Konstante zur Lösung hinzu

= \frac{256 x ^{45}}{9} - 384 x^{40} + 2304 x^{35} - 8064 x^{30} + 18144 x^{25} - 27216 x^{20} + 27216 x^{15} - 17496 x^{10} + 6561 x^{5} + C

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

\int (- 9 x^{2} - 4 x + 1) d x

\int (cos (5 x) - sec^{2} (x)) d x

\int \frac{x - 3}{x ^{4} + 9 x ^{2}} d x

\int x - 3 x d x