integral of ((x^3-x))/(sqrt(x^2+1))

解答

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

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

求解步骤

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

乘开

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

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

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

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

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

化简

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

解答补常数

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

\int x^{2} 4^{- x} d x

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