integral of x^{-3}arctan(x)

解答

\int x^{- 3} arctan (x) d x

- \frac{arctan ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{1}{x} - arctan (x)) + C

求解步骤

\int x^{- 3} arctan (x) d x

化简

x^{- 3} arctan (x) : \frac{arctan ( x )}{x ^{3}}

= \int \frac{arctan ( x )}{x ^{3}} d x

使用分布积分法

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

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

= - \frac{arctan ( x )}{2 x ^{2}} - (- \frac{1}{2} (- \frac{1}{x} - arctan (x)))

化简

= - \frac{arctan ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{1}{x} - arctan (x))

解答补常数

= - \frac{arctan ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{1}{x} - arctan (x)) + C

\int \frac{4}{x ( ln ( x ) ) ^{4}} d x

\int 1 + y^{4} d y

\int \frac{( ln ( x ) ) ^{3}}{x ^{2}} d x

\int \frac{e ^{3 x}}{e ^{2 x} + 1} d x

\int 1 0^{3 x} d x