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人気のある微分積分 >

integral of ((sqrt(x)+2)^8)/(2sqrt(x))

解

\int \frac{( x + 2 ) ^{8}}{2 x} d x

解

\frac{1}{9} (x^{\frac{9}{2}} + 18 x^{4} + 144 x^{\frac{7}{2}} + 672 x^{3} + 2016 x^{\frac{5}{2}} + 4032 x^{2} + 5376 x^{\frac{3}{2}} + 4608 x + 2304 x) + C

解答ステップ

\int \frac{( x + 2 ) ^{8}}{2 x} d x

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{( x + 2 ) ^{8}}{x} d x

拡張

\frac{( x + 2 ) ^{8}}{x} : x^{\frac{7}{2}} + 16 x^{3} + 112 x^{\frac{5}{2}} + 448 x^{2} + 1120 x^{\frac{3}{2}} + 1792 x + 1792 x + 1024 + \frac{256}{x}

= \frac{1}{2} \cdot \int x^{\frac{7}{2}} + 16 x^{3} + 112 x^{\frac{5}{2}} + 448 x^{2} + 1120 x^{\frac{3}{2}} + 1792 x + 1792 x + 1024 + \frac{256}{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

= \frac{1}{2} (\int x^{\frac{7}{2}} d x + \int 16 x^{3} d x + \int 112 x^{\frac{5}{2}} d x + \int 448 x^{2} d x + \int 1120 x^{\frac{3}{2}} d x + \int 1792 x d x + \int 1792 x d x + \int 1024 d x + \int \frac{256}{x} d x)

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 16 x^{3} d x = 4 x^{4}

\int 112 x^{\frac{5}{2}} d x = 32 x^{\frac{7}{2}}

\int 448 x^{2} d x = \frac{448 x ^{3}}{3}

\int 1120 x^{\frac{3}{2}} d x = 448 x^{\frac{5}{2}}

\int 1792 x d x = 896 x^{2}

\int 1792 x d x = \frac{3584}{3} x^{\frac{3}{2}}

\int 1024 d x = 1024 x

\int \frac{256}{x} d x = 512 x

= \frac{1}{2} (\frac{2}{9} x^{\frac{9}{2}} + 4 x^{4} + 32 x^{\frac{7}{2}} + \frac{448 x ^{3}}{3} + 448 x^{\frac{5}{2}} + 896 x^{2} + \frac{3584}{3} x^{\frac{3}{2}} + 1024 x + 512 x)

簡素化

\frac{1}{2} (\frac{2}{9} x^{\frac{9}{2}} + 4 x^{4} + 32 x^{\frac{7}{2}} + \frac{448 x ^{3}}{3} + 448 x^{\frac{5}{2}} + 896 x^{2} + \frac{3584}{3} x^{\frac{3}{2}} + 1024 x + 512 x) : \frac{1}{9} (x^{\frac{9}{2}} + 18 x^{4} + 144 x^{\frac{7}{2}} + 672 x^{3} + 2016 x^{\frac{5}{2}} + 4032 x^{2} + 5376 x^{\frac{3}{2}} + 4608 x + 2304 x)

= \frac{1}{9} (x^{\frac{9}{2}} + 18 x^{4} + 144 x^{\frac{7}{2}} + 672 x^{3} + 2016 x^{\frac{5}{2}} + 4032 x^{2} + 5376 x^{\frac{3}{2}} + 4608 x + 2304 x)

定数を解答に追加する

= \frac{1}{9} (x^{\frac{9}{2}} + 18 x^{4} + 144 x^{\frac{7}{2}} + 672 x^{3} + 2016 x^{\frac{5}{2}} + 4032 x^{2} + 5376 x^{\frac{3}{2}} + 4608 x + 2304 x) + C

グラフ

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