integral of (x^2+7)(sqrt(x)-7)

解答

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

\frac{2}{7} x^{\frac{7}{2}} - \frac{7 x ^{3}}{3} + \frac{14}{3} x^{\frac{3}{2}} - 49 x + C

求解步骤

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

乘开

(x^{2} + 7) (x - 7) : x^{\frac{5}{2}} - 7 x^{2} + 7 x - 49

= \int x^{\frac{5}{2}} - 7 x^{2} + 7 x - 49 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 7 x^{2} d x + \int 7 x d x - \int 49 d x

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

\int 7 x^{2} d x = \frac{7 x ^{3}}{3}

\int 7 x d x = \frac{14}{3} x^{\frac{3}{2}}

\int 49 d x = 49 x

= \frac{2}{7} x^{\frac{7}{2}} - \frac{7 x ^{3}}{3} + \frac{14}{3} x^{\frac{3}{2}} - 49 x

解答补常数

= \frac{2}{7} x^{\frac{7}{2}} - \frac{7 x ^{3}}{3} + \frac{14}{3} x^{\frac{3}{2}} - 49 x + C

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

\int u ln (u) d u

\int 2 x x - 4 d x

\int \frac{3 x - 1}{x ^{2} - 2 x + 10} d x

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