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人気のある微分積分 >

integral of (sec(3u)+csc(3u))^2

解

\int (sec (3 u) + csc (3 u))^{2} d u

解

\frac{1}{3} (tan (3 u) + ln (tan^{2} (3 u)) - cot (3 u)) + C

解答ステップ

\int (sec (3 u) + csc (3 u))^{2} d u

拡張

(sec (3 u) + csc (3 u))^{2} : sec^{2} (3 u) + 2 sec (3 u) csc (3 u) + csc^{2} (3 u)

= \int sec^{2} (3 u) + 2 sec (3 u) csc (3 u) + csc^{2} (3 u) d u

u置換積分法を適用する

= \int (sec^{2} (v) + 2 sec (v) csc (v) + csc^{2} (v)) \frac{1}{3} d v

定数を除く:

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

= \frac{1}{3} \cdot \int sec^{2} (v) + 2 sec (v) csc (v) + csc^{2} (v) d v

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}{3} (\int sec^{2} (v) d v + \int 2 sec (v) csc (v) d v + \int csc^{2} (v) d v)

\int sec^{2} (v) d v = tan (v)

\int 2 sec (v) csc (v) d v = ln (tan^{2} (v))

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

= \frac{1}{3} (tan (v) + ln (tan^{2} (v)) - cot (v))

代用を戻す

v = 3 u

= \frac{1}{3} (tan (3 u) + ln (tan^{2} (3 u)) - cot (3 u))

定数を解答に追加する

= \frac{1}{3} (tan (3 u) + ln (tan^{2} (3 u)) - cot (3 u)) + C

グラフ

人気の例

integral of (-e^2)/x+sqrt(3)*3^x

\int \frac{- e ^{2}}{x} + 3 \cdot 3^{x} d x

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\int cos (3 θ - 1) d θ

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\int (\frac{x ^{3} + 1}{x ^{5}}) d x

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\int ln (w) d w

integral of (8-2x)^2

\int (8 - 2 x)^{2} d x