integral of ((a+bx^2))/(sqrt(3ax+bx^3))

解答

\int \frac{( a + b x ^{2} )}{3 a x + b x ^{3}} d x

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

求解步骤

\int \frac{a + b x ^{2}}{3 a x + b x ^{3}} d x

使用换元积分法

= \int \frac{1}{3 u} d u

提出常数:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

使用幂法则

= \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

u = 3 a x + b x^{3}

代回

= \frac{1}{3} \cdot 2 (3 a x + b x^{3})^{\frac{1}{2}}

化简

\frac{1}{3} \cdot 2 (3 a x + b x^{3})^{\frac{1}{2}} : \frac{2}{3} 3 a x + b x^{3}

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

解答补常数

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