integral of (y^2-1/(y^2))^3

解

\int (y^{2} - \frac{1}{y ^{2}})^{3} d y

\frac{y ^{7}}{7} - y^{3} - \frac{3}{y} + \frac{1}{5 y ^{5}} + C

解答ステップ

\int (y^{2} - \frac{1}{y ^{2}})^{3} d y

拡張

(y^{2} - \frac{1}{y ^{2}})^{3} : y^{6} - 3 y^{2} + \frac{3}{y ^{2}} - \frac{1}{y ^{6}}

= \int y^{6} - 3 y^{2} + \frac{3}{y ^{2}} - \frac{1}{y ^{6}} d y

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

= \int y^{6} d y - \int 3 y^{2} d y + \int \frac{3}{y ^{2}} d y - \int \frac{1}{y ^{6}} d y

\int y^{6} d y = \frac{y ^{7}}{7}

\int 3 y^{2} d y = y^{3}

\int \frac{3}{y ^{2}} d y = - \frac{3}{y}

\int \frac{1}{y ^{6}} d y = - \frac{1}{5 y ^{5}}

= \frac{y ^{7}}{7} - y^{3} - \frac{3}{y} - (- \frac{1}{5 y ^{5}})

簡素化

= \frac{y ^{7}}{7} - y^{3} - \frac{3}{y} + \frac{1}{5 y ^{5}}

定数を解答に追加する

= \frac{y ^{7}}{7} - y^{3} - \frac{3}{y} + \frac{1}{5 y ^{5}} + C