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人気のある微分積分 >

integral from 0 to (5pi)/4 of sec^4(t/5)

解

\int_{0}^{\frac{5 π}{4}} sec^{4} (\frac{t}{5}) d t

解

\frac{20}{3}

+1

十進法表記

6.66666 \dots

解答ステップ

\int_{0}^{\frac{5 π}{4}} sec^{4} (\frac{t}{5}) d t

u置換積分法を適用する

= \int_{0}^{\frac{π}{4}} sec^{4} (u) \cdot 5 d u

定数を除く:

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

= 5 \cdot \int_{0}^{\frac{π}{4}} sec^{4} (u) d u

三角関数の公式を使用して書き換える

= 5 \cdot \int_{0}^{\frac{π}{4}} (1 + tan^{2} (u)) sec^{2} (u) d u

u置換積分法を適用する

= 5 \cdot \int_{0}^{1} 1 + v^{2} d v

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

= 5 (\int_{0}^{1} 1 d v + \int_{0}^{1} v^{2} d v)

\int_{0}^{1} 1 d v = 1

\int_{0}^{1} v^{2} d v = \frac{1}{3}

= 5 (1 + \frac{1}{3})

簡素化

5 (1 + \frac{1}{3}) : \frac{20}{3}

= \frac{20}{3}

グラフ

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