integral of x/(sqrt(25+3x))

解答

\int \frac{x}{25 + 3 x} d x

\frac{2}{9} (\frac{1}{3} (25 + 3 x)^{\frac{3}{2}} - 25 25 + 3 x) + C

求解步骤

\int \frac{x}{25 + 3 x} d x

使用换元积分法

= \int \frac{2 ( u ^{2} - 25 )}{9} d u

提出常数:

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

= \frac{2}{9} \cdot \int u^{2} - 25 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

= \frac{2}{9} (\int u^{2} d u - \int 25 d u)

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

\int 25 d u = 25 u

= \frac{2}{9} (\frac{u ^{3}}{3} - 25 u)

u = 25 + 3 x

代回

= \frac{2}{9} (\frac{( 25 + 3 x ) ^{3}}{3} - 25 25 + 3 x)

化简

\frac{2}{9} (\frac{( 25 + 3 x ) ^{3}}{3} - 25 25 + 3 x) : \frac{2}{9} (\frac{1}{3} (25 + 3 x)^{\frac{3}{2}} - 25 25 + 3 x)

= \frac{2}{9} (\frac{1}{3} (25 + 3 x)^{\frac{3}{2}} - 25 25 + 3 x)

解答补常数

= \frac{2}{9} (\frac{1}{3} (25 + 3 x)^{\frac{3}{2}} - 25 25 + 3 x) + C