integral of (tan(x)+1/(tan(x)))^2

解

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

- 2 cot (2 x) + C

解答ステップ

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

三角関数の公式を使用して書き換える

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

拡張

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

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

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

\int 2 d x = 2 x

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

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

簡素化

tan (x) - x + 2 x - cot (x) - x : - 2 cot (2 x)

= - 2 cot (2 x)

定数を解答に追加する

= - 2 cot (2 x) + C