integral of 42x(x+7)^5

解

\int 42 x (x + 7)^{5} d x

42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2}) + C

解答ステップ

\int 42 x (x + 7)^{5} d x

定数を除く:

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

= 42 \cdot \int x (x + 7)^{5} d x

拡張

x (x + 7)^{5} : x^{6} + 35 x^{5} + 490 x^{4} + 3430 x^{3} + 12005 x^{2} + 16807 x

= 42 \cdot \int x^{6} + 35 x^{5} + 490 x^{4} + 3430 x^{3} + 12005 x^{2} + 16807 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

= 42 (\int x^{6} d x + \int 35 x^{5} d x + \int 490 x^{4} d x + \int 3430 x^{3} d x + \int 12005 x^{2} d x + \int 16807 x d x)

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

\int 35 x^{5} d x = \frac{35 x ^{6}}{6}

\int 490 x^{4} d x = 98 x^{5}

\int 3430 x^{3} d x = \frac{1715 x ^{4}}{2}

\int 12005 x^{2} d x = \frac{12005 x ^{3}}{3}

\int 16807 x d x = \frac{16807 x ^{2}}{2}

= 42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2})

定数を解答に追加する

= 42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2}) + C