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人気のある微分積分 >

integral of (csc^6(2x))/(sqrt(cot(2x)))

解

\int \frac{csc ^{6} ( 2 x )}{cot ( 2 x )} d x

解

- cot (2 x) - \frac{2}{5} cot^{\frac{5}{2}} (2 x) - \frac{1}{9} cot^{\frac{9}{2}} (2 x) + C

解答ステップ

\int \frac{csc ^{6} ( 2 x )}{cot ( 2 x )} d x

u置換積分法を適用する

= \int - 1 - 2 u^{4} - u^{8} 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

= - \int 1 d u - \int 2 u^{4} d u - \int u^{8} d u

\int 1 d u = u

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

\int u^{8} d u = \frac{u ^{9}}{9}

= - u - \frac{2 u ^{5}}{5} - \frac{u ^{9}}{9}

代用を戻す

u = cot (2 x)

= - cot (2 x) - \frac{2 ( cot ( 2 x ) ) ^{5}}{5} - \frac{( cot ( 2 x ) ) ^{9}}{9}

簡素化

- cot (2 x) - \frac{2 ( cot ( 2 x ) ) ^{5}}{5} - \frac{( cot ( 2 x ) ) ^{9}}{9} : - cot (2 x) - \frac{2}{5} cot^{\frac{5}{2}} (2 x) - \frac{1}{9} cot^{\frac{9}{2}} (2 x)

= - cot (2 x) - \frac{2}{5} cot^{\frac{5}{2}} (2 x) - \frac{1}{9} cot^{\frac{9}{2}} (2 x)

定数を解答に追加する

= - cot (2 x) - \frac{2}{5} cot^{\frac{5}{2}} (2 x) - \frac{1}{9} cot^{\frac{9}{2}} (2 x) + C

グラフ

人気の例

integral of (x^2+4)/(x(x-1)^2)

\int \frac{x ^{2} + 4}{x ( x - 1 ) ^{2}} d x

integral of 1/2 tcos(t^2)

\int \frac{1}{2} t cos (t^{2}) d t

integral of xln(3x^3)

\int x ln (3 x^{3}) d x

integral of 4sec^2(8x+3)

\int 4 sec^{2} (8 x + 3) d x

integral of 7tan(x)sec^5(x)

\int 7 tan (x) sec^{5} (x) d x