integral of ((4+x^7)^67x^6)

Lösung

\int ((4 + x^{7})^{6} 7 x^{6}) d x

4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7} + C

Schritte zur Lösung

\int (4 + x^{7})^{6} \cdot 7 x^{6} d x

Entferne die Konstante:

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

= 7 \cdot \int (4 + x^{7})^{6} x^{6} d x

Multipliziere aus

(4 + x^{7})^{6} x^{6} : 4096 x^{6} + 6144 x^{13} + 3840 x^{20} + 1280 x^{27} + 240 x^{34} + 24 x^{41} + x^{48}

= 7 \cdot \int 4096 x^{6} + 6144 x^{13} + 3840 x^{20} + 1280 x^{27} + 240 x^{34} + 24 x^{41} + x^{48} 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

= 7 (\int 4096 x^{6} d x + \int 6144 x^{13} d x + \int 3840 x^{20} d x + \int 1280 x^{27} d x + \int 240 x^{34} d x + \int 24 x^{41} d x + \int x^{48} d x)

\int 4096 x^{6} d x = \frac{4096 x ^{7}}{7}

\int 6144 x^{13} d x = \frac{3072 x ^{14}}{7}

\int 3840 x^{20} d x = \frac{1280 x ^{21}}{7}

\int 1280 x^{27} d x = \frac{320 x ^{28}}{7}

\int 240 x^{34} d x = \frac{48 x ^{35}}{7}

\int 24 x^{41} d x = \frac{4 x ^{42}}{7}

\int x^{48} d x = \frac{x ^{49}}{49}

= 7 (\frac{4096 x ^{7}}{7} + \frac{3072 x ^{14}}{7} + \frac{1280 x ^{21}}{7} + \frac{320 x ^{28}}{7} + \frac{48 x ^{35}}{7} + \frac{4 x ^{42}}{7} + \frac{x ^{49}}{49})

Vereinfache

7 (\frac{4096 x ^{7}}{7} + \frac{3072 x ^{14}}{7} + \frac{1280 x ^{21}}{7} + \frac{320 x ^{28}}{7} + \frac{48 x ^{35}}{7} + \frac{4 x ^{42}}{7} + \frac{x ^{49}}{49}) : 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7}

= 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7}

Füge eine Konstante zur Lösung hinzu

= 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7} + C

x \to 3 lim (10)

\frac{\partial}{\partial z} (ln (x yz))

d er i v a t i v e 3 x + 8

\int e^{c o s (37 t)} sin (37 t) d t

x \to - 9 lim (\frac{x + 8}{x + 9})