integral of (sec(x)+csc(x))^2

解

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

tan (x) + 2 ln ∣ tan (x) ∣ - cot (x) + C

解答ステップ

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

三角関数の公式を使用して書き換える

= \int \frac{( sin ( x ) + cos ( x ) ) ^{2}}{cos ^{2} ( x ) sin ^{2} ( x )} d x

拡張

\frac{( sin ( x ) + cos ( x ) ) ^{2}}{cos ^{2} ( x ) sin ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} + \frac{2}{cos ( x ) sin ( x )} + \frac{1}{sin ^{2} ( x )}

= \int \frac{1}{cos ^{2} ( x )} + \frac{2}{cos ( x ) sin ( x )} + \frac{1}{sin ^{2} ( 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}{cos ^{2} ( x )} d x + \int \frac{2}{cos ( x ) sin ( x )} d x + \int \frac{1}{sin ^{2} ( x )} d x

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

\int \frac{2}{cos ( x ) sin ( x )} d x = 2 ln ∣ tan (x) ∣

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

= tan (x) + 2 ln ∣ tan (x) ∣ - cot (x)

定数を解答に追加する

= tan (x) + 2 ln ∣ tan (x) ∣ - cot (x) + C