integral of 4x(2x^2+1)^8

Lösung

\int 4 x (2 x^{2} + 1)^{8} d x

4 (\frac{128 x ^{18}}{9} + 64 x^{16} + 128 x^{14} + \frac{448 x ^{12}}{3} + 112 x^{10} + 56 x^{8} + \frac{56 x ^{6}}{3} + 4 x^{4} + \frac{x ^{2}}{2}) + C

Schritte zur Lösung

\int 4 x (2 x^{2} + 1)^{8} d x

Entferne die Konstante:

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

= 4 \cdot \int x (2 x^{2} + 1)^{8} d x

Multipliziere aus

x (2 x^{2} + 1)^{8} : 256 x^{17} + 1024 x^{15} + 1792 x^{13} + 1792 x^{11} + 1120 x^{9} + 448 x^{7} + 112 x^{5} + 16 x^{3} + x

= 4 \cdot \int 256 x^{17} + 1024 x^{15} + 1792 x^{13} + 1792 x^{11} + 1120 x^{9} + 448 x^{7} + 112 x^{5} + 16 x^{3} + x 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

= 4 (\int 256 x^{17} d x + \int 1024 x^{15} d x + \int 1792 x^{13} d x + \int 1792 x^{11} d x + \int 1120 x^{9} d x + \int 448 x^{7} d x + \int 112 x^{5} d x + \int 16 x^{3} d x + \int x d x)

\int 256 x^{17} d x = \frac{128 x ^{18}}{9}

\int 1024 x^{15} d x = 64 x^{16}

\int 1792 x^{13} d x = 128 x^{14}

\int 1792 x^{11} d x = \frac{448 x ^{12}}{3}

\int 1120 x^{9} d x = 112 x^{10}

\int 448 x^{7} d x = 56 x^{8}

\int 112 x^{5} d x = \frac{56 x ^{6}}{3}

\int 16 x^{3} d x = 4 x^{4}

\int x d x = \frac{x ^{2}}{2}

= 4 (\frac{128 x ^{18}}{9} + 64 x^{16} + 128 x^{14} + \frac{448 x ^{12}}{3} + 112 x^{10} + 56 x^{8} + \frac{56 x ^{6}}{3} + 4 x^{4} + \frac{x ^{2}}{2})

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{128 x ^{18}}{9} + 64 x^{16} + 128 x^{14} + \frac{448 x ^{12}}{3} + 112 x^{10} + 56 x^{8} + \frac{56 x ^{6}}{3} + 4 x^{4} + \frac{x ^{2}}{2}) + C

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

t a y l or x^{3}, at \frac{16}{3}

y^{^{''}} - 4 y^{^{'}} - 5 y = 0, y (1) = 0, y^{^{'}} (1) = 8

d er i v a t i v e - cot (x)

y^{^{''''}} + 2 y^{^{''}} = 0