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人気のある微分積分 >

integral from 0 to 1 of x^2sqrt(x^2+3)

解

\int_{0}^{1} x^{2} x^{2} + 3 d x

解

9 (\frac{5}{36} - \frac{1}{16} ln (3))

+1

十進法表記

0.63203 \dots

解答ステップ

\int_{0}^{1} x^{2} x^{2} + 3 d x

三角関数による置換を適用する

= \int_{0}^{\frac{π}{6}} 9 sec^{3} (u) tan^{2} (u) d u

定数を除く:

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

= 9 \cdot \int_{0}^{\frac{π}{6}} sec^{3} (u) tan^{2} (u) d u

三角関数の公式を使用して書き換える

= 9 \cdot \int_{0}^{\frac{π}{6}} sec^{3} (u) (- 1 + sec^{2} (u)) d u

拡張

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= 9 \cdot \int_{0}^{\frac{π}{6}} - sec^{3} (u) + sec^{5} (u) 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

= 9 (- \int_{0}^{\frac{π}{6}} sec^{3} (u) d u + \int_{0}^{\frac{π}{6}} sec^{5} (u) d u)

\int_{0}^{\frac{π}{6}} sec^{3} (u) d u = \frac{1}{3} + \frac{1}{4} ln (3)

\int_{0}^{\frac{π}{6}} sec^{5} (u) d u = \frac{17}{36} + \frac{3}{16} ln (3)

= 9 (- (\frac{1}{3} + \frac{1}{4} ln (3)) + \frac{17}{36} + \frac{3}{16} ln (3))

拡張

- (\frac{1}{3} + \frac{1}{4} ln (3)) + \frac{17}{36} + \frac{3}{16} ln (3) : \frac{5}{36} - \frac{1}{16} ln (3)

= 9 (\frac{5}{36} - \frac{1}{16} ln (3))

グラフ

人気の例

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