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人気のある微分積分 >

integral of 1/(z^3sqrt(z^2-81))

解

\int \frac{1}{z ^{3} z ^{2} - 81} d z

解

\frac{1}{1458} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z))) + C

解答ステップ

\int \frac{1}{z ^{3} z ^{2} - 81} d z

三角関数による置換を適用する

= \int \frac{1}{729 sec ^{2} ( u )} d u

定数を除く:

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

= \frac{1}{729} \cdot \int \frac{1}{sec ^{2} ( u )} d u

三角関数の公式を使用して書き換える

= \frac{1}{729} \cdot \int \frac{1 + cos ( 2 u )}{2} d u

定数を除く:

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

= \frac{1}{729} \cdot \frac{1}{2} \cdot \int 1 + cos (2 u) d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{729} \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{729} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

代用を戻す

u = arcsec (\frac{1}{9} z)

= \frac{1}{729} \cdot \frac{1}{2} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z)))

簡素化

\frac{1}{729} \cdot \frac{1}{2} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z))) : \frac{1}{1458} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z)))

= \frac{1}{1458} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z)))

定数を解答に追加する

= \frac{1}{1458} (arcsec (\frac{1}{9} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{9} z))) + C

グラフ

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