integral of (2sech^2(t))/(tanh(t))

解

\int \frac{2 sech ^{2} ( t )}{tanh ( t )} d t

2 (ln ∣ sinh (t) ∣ - ln ∣ cosh (t) ∣) + C

解答ステップ

\int \frac{2 sech ^{2} ( t )}{tanh ( t )} d t

定数を除く:

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

= 2 \cdot \int \frac{sech ^{2} ( t )}{tanh ( t )} d t

三角関数の公式を使用して書き換える

= 2 \cdot \int \frac{1 - tanh ^{2} ( t )}{tanh ( t )} d t

拡張

\frac{1 - tanh ^{2} ( t )}{tanh ( t )} : \frac{1}{tanh ( t )} - tanh (t)

= 2 \cdot \int \frac{1}{tanh ( t )} - tanh (t) d t

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

= 2 (\int \frac{1}{tanh ( t )} d t - \int tanh (t) d t)

\int \frac{1}{tanh ( t )} d t = ln ∣ sinh (t) ∣

\int tanh (t) d t = ln ∣ cosh (t) ∣

= 2 (ln ∣ sinh (t) ∣ - ln ∣ cosh (t) ∣)

定数を解答に追加する

= 2 (ln ∣ sinh (t) ∣ - ln ∣ cosh (t) ∣) + C