integral of 3xsqrt(x+3)

解答

\int 3 x x + 3 d x

3 (\frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}}) + C

求解步骤

\int 3 x x + 3 d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x x + 3 d x

使用换元积分法

= 3 \cdot \int (u - 3) u d u

乘开

(u - 3) u : u^{\frac{3}{2}} - 3 u

= 3 \cdot \int u^{\frac{3}{2}} - 3 u d u

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

= 3 (\int u^{\frac{3}{2}} d u - \int 3 u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 3 u d u = 2 u^{\frac{3}{2}}

= 3 (\frac{2}{5} u^{\frac{5}{2}} - 2 u^{\frac{3}{2}})

u = x + 3

代回

= 3 (\frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}})

解答补常数

= 3 (\frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}}) + C

\int_{0}^{4} \frac{1}{2 x + 1} d x

x \to - \infty lim (9 x^{3})

\frac{d}{d x} (\frac{m}{2 π ( a - x )})

\int_{0}^{\frac{π}{3}} 1

\frac{d}{d x} ((ln (x)) (ln (3 x - 7)))