integral of (xcot^2(x^2))/(csc(x^2))

解

\int \frac{x cot ^{2} ( x ^{2} )}{csc ( x ^{2} )} d x

\frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2}) + C

解答ステップ

\int \frac{x cot ^{2} ( x ^{2} )}{csc ( x ^{2} )} d x

三角関数の公式を使用して書き換える

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

拡張

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

= \int - \frac{x}{csc ( x ^{2} )} + x csc (x^{2}) 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{x}{csc ( x ^{2} )} d x + \int x csc (x^{2}) d x

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

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

= - (- \frac{1}{2} cos (x^{2})) + \frac{1}{2} ln tan (\frac{x ^{2}}{2})

簡素化

= \frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2})

定数を解答に追加する

= \frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2}) + C