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

解

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

ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣ + C

解答ステップ

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

三角関数の公式を使用して書き換える

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

拡張

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

= \int \frac{1}{tanh ( x )} - tanh (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

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

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

\int tanh (x) d x = ln ∣ cosh (x) ∣

= ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣

定数を解答に追加する

= ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣ + C