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

解答

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

- \frac{1}{x} - \frac{1}{x ^{2}} - \frac{1}{3 x ^{3}} + C

求解步骤

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

化简

(1 + \frac{1}{x})^{2} \frac{1}{x ^{2}} : \frac{( x + 1 ) ^{2}}{x ^{4}}

= \int \frac{( x + 1 ) ^{2}}{x ^{4}} d x

乘开

\frac{( x + 1 ) ^{2}}{x ^{4}} : \frac{1}{x ^{2}} + \frac{2}{x ^{3}} + \frac{1}{x ^{4}}

= \int \frac{1}{x ^{2}} + \frac{2}{x ^{3}} + \frac{1}{x ^{4}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{2}} d x + \int \frac{2}{x ^{3}} d x + \int \frac{1}{x ^{4}} d x

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

\int \frac{2}{x ^{3}} d x = - \frac{1}{x ^{2}}

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

= - \frac{1}{x} - \frac{1}{x ^{2}} - \frac{1}{3 x ^{3}}

解答补常数

= - \frac{1}{x} - \frac{1}{x ^{2}} - \frac{1}{3 x ^{3}} + C

\int (cos (x) tan (x)) d x

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

\int 6 (e^{4 - 3 x})^{2} d x

\int (e^{3 x} + e^{- x})^{2} e^{x + 2} d x

\int \frac{y}{( y + 5 ) ^{2} ( y - 5 )} d y