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人気のある微分積分 >

integral from 0 to pi/3 of (4sin(θ))/(1-4cos(θ))

解

\int_{0}^{\frac{π}{3}} \frac{4 sin ( θ )}{1 - 4 cos ( θ )} d θ

解

- ln (3)

+1

十進法表記

- 1.09861 \dots

解答ステップ

\int_{0}^{\frac{π}{3}} \frac{4 sin ( θ )}{1 - 4 cos ( θ )} d θ

定数を除く:

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

= 4 \cdot \int_{0}^{\frac{π}{3}} \frac{sin ( θ )}{1 - 4 cos ( θ )} d θ

u置換積分法を適用する

= 4 \cdot \int_{- 3}^{- 1} \frac{1}{4 u} d u

定数を除く:

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

= 4 \cdot \frac{1}{4} \cdot \int_{- 3}^{- 1} \frac{1}{u} d u

共通積分を使用する:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{4} [ln ∣ u ∣]_{- 3}^{- 1}

簡素化

4 \cdot \frac{1}{4} [ln ∣ u ∣]_{- 3}^{- 1} : [ln ∣ u ∣]_{- 3}^{- 1}

= [ln ∣ u ∣]_{- 3}^{- 1}

限度を計算する:

- ln (3)

= - ln (3)

グラフ

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