integral of xos(8x^2+pi)

解

\int x os (8 x^{2} + π) d x

os (\frac{π x ^{2}}{2} + 2 x^{4}) + C

解答ステップ

\int x os (8 x^{2} + π) d x

定数を除く:

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

= os \cdot \int (π + 8 x^{2}) x d x

拡張

(π + 8 x^{2}) x : π x + 8 x^{3}

= os \cdot \int π x + 8 x^{3} 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

= os (\int π x d x + \int 8 x^{3} d x)

\int π x d x = \frac{π x ^{2}}{2}

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

= os (\frac{π x ^{2}}{2} + 2 x^{4})

定数を解答に追加する

= os (\frac{π x ^{2}}{2} + 2 x^{4}) + C