integral of (3x^2+10x)(x^3+5x^2+6)

解

\int (3 x^{2} + 10 x) (x^{3} + 5 x^{2} + 6) d x

\frac{x ^{6}}{2} + 5 x^{5} + \frac{25 x ^{4}}{2} + 6 x^{3} + 30 x^{2} + C

解答ステップ

\int (3 x^{2} + 10 x) (x^{3} + 5 x^{2} + 6) d x

拡張

(3 x^{2} + 10 x) (x^{3} + 5 x^{2} + 6) : 3 x^{5} + 25 x^{4} + 50 x^{3} + 18 x^{2} + 60 x

= \int 3 x^{5} + 25 x^{4} + 50 x^{3} + 18 x^{2} + 60 x 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

= \int 3 x^{5} d x + \int 25 x^{4} d x + \int 50 x^{3} d x + \int 18 x^{2} d x + \int 60 x d x

\int 3 x^{5} d x = \frac{x ^{6}}{2}

\int 25 x^{4} d x = 5 x^{5}

\int 50 x^{3} d x = \frac{25 x ^{4}}{2}

\int 18 x^{2} d x = 6 x^{3}

\int 60 x d x = 30 x^{2}

= \frac{x ^{6}}{2} + 5 x^{5} + \frac{25 x ^{4}}{2} + 6 x^{3} + 30 x^{2}

定数を解答に追加する

= \frac{x ^{6}}{2} + 5 x^{5} + \frac{25 x ^{4}}{2} + 6 x^{3} + 30 x^{2} + C