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

解答

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

\frac{2}{7} x^{\frac{7}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C

求解步骤

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

乘开

(x^{2} + 2) x : x^{\frac{5}{2}} + 2 x

= \int x^{\frac{5}{2}} + 2 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 x^{\frac{5}{2}} d x + \int 2 x d x

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

= \frac{2}{7} x^{\frac{7}{2}} + \frac{4}{3} x^{\frac{3}{2}}

解答补常数

= \frac{2}{7} x^{\frac{7}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C

\int \frac{1 + cos ( x )}{sin ( x )} d x

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