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人気のある微分積分 >

integral of 9tan^2(x)sec(x)

解

\int 9 tan^{2} (x) sec (x) d x

解

9 (\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

解答ステップ

\int 9 tan^{2} (x) sec (x) d x

定数を除く:

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

= 9 \cdot \int tan^{2} (x) sec (x) d x

三角関数の公式を使用して書き換える

= 9 \cdot \int (- 1 + sec^{2} (x)) sec (x) d x

拡張

(- 1 + sec^{2} (x)) sec (x) : - sec (x) + sec^{3} (x)

= 9 \cdot \int - sec (x) + sec^{3} (x) d x

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

= 9 (- \int sec (x) d x + \int sec^{3} (x) d x)

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

\int sec^{3} (x) d x = \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= 9 (- ln ∣ tan (x) + sec (x) ∣ + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

簡素化

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

= 9 (\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

定数を解答に追加する

= 9 (\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

グラフ

人気の例

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integral of (2x^2)/(20-5x^3)

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