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人気のある微分積分 >

integral of (tan(4x)+sec(4x))^2

解

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

解

\frac{1}{2} (- 2 x + tan (4 x) + sec (4 x)) + C

解答ステップ

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

u置換積分法を適用する

= \int (tan (u) + sec (u))^{2} \frac{1}{4} d u

定数を除く:

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

= \frac{1}{4} \cdot \int (tan (u) + sec (u))^{2} d u

拡張

(tan (u) + sec (u))^{2} : tan^{2} (u) + 2 tan (u) sec (u) + sec^{2} (u)

= \frac{1}{4} \cdot \int tan^{2} (u) + 2 tan (u) sec (u) + sec^{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}{4} (\int tan^{2} (u) d u + \int 2 tan (u) sec (u) d u + \int sec^{2} (u) d u)

\int tan^{2} (u) d u = - u + tan (u)

\int 2 tan (u) sec (u) d u = 2 sec (u)

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

= \frac{1}{4} (- u + tan (u) + 2 sec (u) + tan (u))

代用を戻す

u = 4 x

= \frac{1}{4} (- 4 x + tan (4 x) + 2 sec (4 x) + tan (4 x))

簡素化

\frac{1}{4} (- 4 x + tan (4 x) + 2 sec (4 x) + tan (4 x)) : \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x))

= \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x))

定数を解答に追加する

= \frac{1}{2} (- 2 x + tan (4 x) + sec (4 x)) + C

グラフ

人気の例

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