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人気のある微分積分 >

integral of (81)/(x^2sqrt(81-x^2))

解

\int \frac{81}{x ^{2} 81 - x ^{2}} d x

解

- \frac{81 - x ^{2}}{x} + C

解答ステップ

\int \frac{81}{x ^{2} 81 - x ^{2}} d x

定数を除く:

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

= 81 \cdot \int \frac{1}{x ^{2} 81 - x ^{2}} d x

三角関数による置換を適用する

= 81 \cdot \int \frac{cot ( u )}{81 cos ( u ) sin ( u )} d u

定数を除く:

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

= 81 \cdot \frac{1}{81} \cdot \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

簡素化

\frac{cot ( u )}{cos ( u ) sin ( u )} : csc^{2} (u)

= 81 \cdot \frac{1}{81} \cdot \int csc^{2} (u) d u

共通積分を使用する:

\int csc^{2} (u) d u = - cot (u)

= 81 \cdot \frac{1}{81} (- cot (u))

代用を戻す

u = arcsin (\frac{1}{9} x)

= 81 \cdot \frac{1}{81} (- cot (arcsin (\frac{1}{9} x)))

簡素化

81 \cdot \frac{1}{81} (- cot (arcsin (\frac{1}{9} x))) : - \frac{81 - x ^{2}}{x}

= - \frac{81 - x ^{2}}{x}

定数を解答に追加する

= - \frac{81 - x ^{2}}{x} + C

グラフ

人気の例

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