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人気のある微分積分 >

integral from 0 to pi/3 of tan^5(x)

解

\int_{0}^{\frac{π}{3}} tan^{5} (x) d x

解

ln (2) + \frac{3}{4}

+1

十進法表記

1.44314 \dots

解答ステップ

\int_{0}^{\frac{π}{3}} tan^{5} (x) d x

三角関数の公式を使用して書き換える

= \int_{0}^{\frac{π}{3}} (- 1 + sec^{2} (x))^{2} tan (x) d x

u置換積分法を適用する

= \int_{1}^{2} \frac{( - 1 + u ^{2} ) ^{2}}{u} d u

拡張

\frac{( - 1 + u ^{2} ) ^{2}}{u} : \frac{1}{u} - 2 u + u^{3}

= \int_{1}^{2} \frac{1}{u} - 2 u + u^{3} 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}^{2} \frac{1}{u} d u - \int_{1}^{2} 2 u d u + \int_{1}^{2} u^{3} d u

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

\int_{1}^{2} 2 u d u = 3

\int_{1}^{2} u^{3} d u = \frac{15}{4}

= ln (2) - 3 + \frac{15}{4}

分数を組み合わせる

- 3 + \frac{15}{4} : \frac{3}{4}

= ln (2) + \frac{3}{4}

グラフ

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