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人気のある微分積分 >

integral from 1 to ln(2) of xe^{4x}

解

\int_{1}^{l n (2)} x e^{4 x} d x

解

- \frac{3 e ^{4}}{16} + 4 ln (2) - 1

+1

十進法表記

- 8.46456 \dots

解答ステップ

\int_{1}^{l n (2)} x e^{4 x} d x

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

= - \int_{l n (2)}^{1} x e^{4 x} d x

u置換積分法を適用する

= - \int_{4 l n (2)}^{4} \frac{e ^{u} u}{16} d u

定数を除く:

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

= - \frac{1}{16} \cdot \int_{4 l n (2)}^{4} e^{u} u d u

部分積分を適用する

= - \frac{1}{16} [e^{u} u - \int e^{u} d u]_{4 l n (2)}^{4}

\int e^{u} d u = e^{u}

= - \frac{1}{16} [e^{u} u - e^{u}]_{4 l n (2)}^{4}

限度を計算する:

3 e^{4} - 64 ln (2) + 16

= - \frac{1}{16} (3 e^{4} - 64 ln (2) + 16)

簡素化

= - \frac{3 e ^{4}}{16} + 4 ln (2) - 1

グラフ

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