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人気のある微分積分 >

integral of tan^2(t)an(t)

解

\int tan^{2} (t) d tan (t)

解

d (- ln ∣ sec (t) ∣ + \frac{sec ^{2} ( t )}{2}) + C

解答ステップ

\int tan^{2} (t) d tan (t) d t

定数を除く:

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

= d \cdot \int tan (t) tan^{2} (t) d t

三角関数の公式を使用して書き換える

= d \cdot \int tan (t) (- 1 + sec^{2} (t)) d t

u置換積分法を適用する

= d \cdot \int \frac{- 1 + u ^{2}}{u} d u

拡張

\frac{- 1 + u ^{2}}{u} : - \frac{1}{u} + u

= d \cdot \int - \frac{1}{u} + 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

= d (- \int \frac{1}{u} d u + \int u d u)

\int \frac{1}{u} d u = ln ∣ u ∣

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

= d (- ln ∣ u ∣ + \frac{u ^{2}}{2})

代用を戻す

u = sec (t)

= d (- ln ∣ sec (t) ∣ + \frac{sec ^{2} ( t )}{2})

定数を解答に追加する

= d (- ln ∣ sec (t) ∣ + \frac{sec ^{2} ( t )}{2}) + C

グラフ

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