integral of (a^2-2x^2)^4*(20x)

解

\int (a^{2} - 2 x^{2})^{4} \cdot (20 x) d x

20 (\frac{a ^{8} x ^{2}}{2} - 2 a^{6} x^{4} + 4 a^{4} x^{6} - 4 a^{2} x^{8} + \frac{8 x ^{10}}{5}) + C

解答ステップ

\int (a^{2} - 2 x^{2})^{4} \cdot 20 x d x

定数を除く:

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

= 20 \cdot \int (a^{2} - 2 x^{2})^{4} x d x

拡張

(a^{2} - 2 x^{2})^{4} x : a^{8} x - 8 a^{6} x^{3} + 24 a^{4} x^{5} - 32 a^{2} x^{7} + 16 x^{9}

= 20 \cdot \int a^{8} x - 8 a^{6} x^{3} + 24 a^{4} x^{5} - 32 a^{2} x^{7} + 16 x^{9} d x

総和規則を適用する:

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

= 20 (\int a^{8} x d x - \int 8 a^{6} x^{3} d x + \int 24 a^{4} x^{5} d x - \int 32 a^{2} x^{7} d x + \int 16 x^{9} d x)

\int a^{8} x d x = \frac{a ^{8} x ^{2}}{2}

\int 8 a^{6} x^{3} d x = 2 a^{6} x^{4}

\int 24 a^{4} x^{5} d x = 4 a^{4} x^{6}

\int 32 a^{2} x^{7} d x = 4 a^{2} x^{8}

\int 16 x^{9} d x = \frac{8 x ^{10}}{5}

= 20 (\frac{a ^{8} x ^{2}}{2} - 2 a^{6} x^{4} + 4 a^{4} x^{6} - 4 a^{2} x^{8} + \frac{8 x ^{10}}{5})

定数を解答に追加する

= 20 (\frac{a ^{8} x ^{2}}{2} - 2 a^{6} x^{4} + 4 a^{4} x^{6} - 4 a^{2} x^{8} + \frac{8 x ^{10}}{5}) + C