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

解

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

\frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x + C

解答ステップ

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

拡張

(3 x^{2} + 5)^{3} : 27 x^{6} + 135 x^{4} + 225 x^{2} + 125

= \int 27 x^{6} + 135 x^{4} + 225 x^{2} + 125 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 27 x^{6} d x + \int 135 x^{4} d x + \int 225 x^{2} d x + \int 125 d x

\int 27 x^{6} d x = \frac{27 x ^{7}}{7}

\int 135 x^{4} d x = 27 x^{5}

\int 225 x^{2} d x = 75 x^{3}

\int 125 d x = 125 x

= \frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x

定数を解答に追加する

= \frac{27 x ^{7}}{7} + 27 x^{5} + 75 x^{3} + 125 x + C