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人気のある微分積分 >

integral from 0 to 5 of x/((1+x)^{3/4)}

解

\int_{0}^{5} \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

解

\frac{24 \cdot 6 ^{\frac{1}{4}} - 4}{5} - 4 \cdot 6^{\frac{1}{4}} + 4

+1

十進法表記

4.45206 \dots

解答ステップ

\int_{0}^{5} \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

u置換積分法を適用する

= \int_{1}^{6} \frac{u - 1}{u ^{\frac{3}{4}}} d u

拡張

\frac{u - 1}{u ^{\frac{3}{4}}} : u^{\frac{1}{4}} - \frac{1}{u ^{\frac{3}{4}}}

= \int_{1}^{6} u^{\frac{1}{4}} - \frac{1}{u ^{\frac{3}{4}}} d u

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_{1}^{6} u^{\frac{1}{4}} d u - \int_{1}^{6} \frac{1}{u ^{\frac{3}{4}}} d u

\int_{1}^{6} u^{\frac{1}{4}} d u = \frac{24 \cdot 6 ^{\frac{1}{4}} - 4}{5}

\int_{1}^{6} \frac{1}{u ^{\frac{3}{4}}} d u = 4 \cdot 6^{\frac{1}{4}} - 4

= \frac{24 \cdot 6 ^{\frac{1}{4}} - 4}{5} - (4 \cdot 6^{\frac{1}{4}} - 4)

簡素化

= \frac{24 \cdot 6 ^{\frac{1}{4}} - 4}{5} - 4 \cdot 6^{\frac{1}{4}} + 4

グラフ

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