integral from 1 to 2 of 9(ln(x))/(x^2)

解

\int_{1}^{2} 9 \frac{ln ( x )}{x ^{2}} d x

9 (\frac{- ln ( 2 ) - 1}{2} + 1)

十進法表記

1.38083 \dots

解答ステップ

\int_{1}^{2} 9 \cdot \frac{ln ( x )}{x ^{2}} d x

定数を除く:

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

= 9 \cdot \int_{1}^{2} \frac{ln ( x )}{x ^{2}} d x

部分積分を適用する

= 9 [- \frac{ln ( x )}{x} - \int - \frac{1}{x ^{2}} d x]_{1}^{2}

\int - \frac{1}{x ^{2}} d x = \frac{1}{x}

= 9 [- \frac{ln ( x )}{x} - \frac{1}{x}]_{1}^{2}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= 9 [\frac{- ln ( x ) - 1}{x}]_{1}^{2}

限度を計算する:

\frac{- ln ( 2 ) - 1}{2} + 1

= 9 (\frac{- ln ( 2 ) - 1}{2} + 1)