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人気のある微分積分 >

integral from-1 to 1 of (3x)/((x+5)^2)

解

\int_{- 1}^{1} \frac{3 x}{( x + 5 ) ^{2}} d x

解

3 (ln (\frac{3}{2}) - \frac{5}{12})

+1

十進法表記

- 0.03360 \dots

解答ステップ

\int_{- 1}^{1} \frac{3 x}{( x + 5 ) ^{2}} d x

定数を除く:

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

= 3 \cdot \int_{- 1}^{1} \frac{x}{( x + 5 ) ^{2}} d x

u置換積分法を適用する

= 3 \cdot \int_{4}^{6} \frac{u - 5}{u ^{2}} d u

拡張

\frac{u - 5}{u ^{2}} : \frac{1}{u} - \frac{5}{u ^{2}}

= 3 \cdot \int_{4}^{6} \frac{1}{u} - \frac{5}{u ^{2}} 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

= 3 (\int_{4}^{6} \frac{1}{u} d u - \int_{4}^{6} \frac{5}{u ^{2}} d u)

\int_{4}^{6} \frac{1}{u} d u = ln (\frac{3}{2})

\int_{4}^{6} \frac{5}{u ^{2}} d u = \frac{5}{12}

= 3 (ln (\frac{3}{2}) - \frac{5}{12})

グラフ

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