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人気のある微分積分 >

integral from pi/6 to pi/3 of tan(x)

解

\int_{\frac{π}{6}}^{\frac{π}{3}} tan (x) d x

解

\frac{1}{2} ln (3)

+1

十進法表記

0.54930 \dots

解答ステップ

\int_{\frac{π}{6}}^{\frac{π}{3}} tan (x) d x

三角関数の公式を使用して書き換える

= \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{sin ( x )}{cos ( x )} d x

u置換積分法を適用する

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

定数を除く:

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

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

共通積分を使用する:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}})

簡素化

= [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}}

限度を計算する:

\frac{1}{2} ln (3)

= \frac{1}{2} ln (3)

グラフ

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