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

解答

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

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

求解步骤

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

乘开

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

= \int a^{2} + 2 a 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 a^{2} d x + \int 2 a x d x + \int x d x

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

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

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

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

解答补常数

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

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