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

解答

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

x + \frac{4}{3} x^{\frac{3}{2}} + \frac{x ^{2}}{2} + C

求解步骤

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

乘开

(1 + x)^{2} : 1 + 2 x + x

= \int 1 + 2 x + x 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 1 d x + \int 2 x d x + \int x d x

\int 1 d x = x

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

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

= x + \frac{4}{3} x^{\frac{3}{2}} + \frac{x ^{2}}{2}

解答补常数

= x + \frac{4}{3} x^{\frac{3}{2}} + \frac{x ^{2}}{2} + C

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

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