integral of (2x^4+9x^2+8)/(x(x^2+2)^2)

解答

\int \frac{2 x ^{4} + 9 x ^{2} + 8}{x ( x ^{2} + 2 ) ^{2}} d x

\frac{4}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x) + C

求解步骤

\int \frac{2 x ^{4} + 9 x ^{2} + 8}{x ( x ^{2} + 2 ) ^{2}} d x

乘开

\frac{2 x ^{4} + 9 x ^{2} + 8}{x ( x ^{2} + 2 ) ^{2}} : \frac{2 x ^{4}}{x ( x ^{2} + 2 ) ^{2}} + \frac{9 x ^{2}}{x ( x ^{2} + 2 ) ^{2}} + \frac{8}{x ( x ^{2} + 2 ) ^{2}}

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{2 x ^{4}}{x ( x ^{2} + 2 ) ^{2}} d x + \int \frac{9 x ^{2}}{x ( x ^{2} + 2 ) ^{2}} d x + \int \frac{8}{x ( x ^{2} + 2 ) ^{2}} d x

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

\int \frac{9 x ^{2}}{x ( x ^{2} + 2 ) ^{2}} d x = - \frac{9}{2 ( x ^{2} + 2 )}

\int \frac{8}{x ( x ^{2} + 2 ) ^{2}} d x = 2 ln (x) - ln x^{2} + 2 + \frac{2}{x ^{2} + 2}

= ln x^{2} + 2 + \frac{2}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x) - ln x^{2} + 2 + \frac{2}{x ^{2} + 2}

化简

ln x^{2} + 2 + \frac{2}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x) - ln x^{2} + 2 + \frac{2}{x ^{2} + 2} : \frac{4}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x)

= \frac{4}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x)

解答补常数

= \frac{4}{x ^{2} + 2} - \frac{9}{2 ( x ^{2} + 2 )} + 2 ln (x) + C

\int (2 x^{3} + 7 3 x^{2}) d x

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