integral of (5x^2+1)sqrt(5x^3+3x-2)

解答

\int (5 x^{2} + 1) 5 x^{3} + 3 x - 2 d x

\frac{2}{9} (5 x^{3} + 3 x - 2)^{\frac{3}{2}} + C

求解步骤

\int (5 x^{2} + 1) 5 x^{3} + 3 x - 2 d x

使用换元积分法

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

使用幂法则

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

u = 5 x^{3} + 3 x - 2

代回

= \frac{1}{3} \cdot \frac{2}{3} (5 x^{3} + 3 x - 2)^{\frac{3}{2}}

化简

\frac{1}{3} \cdot \frac{2}{3} (5 x^{3} + 3 x - 2)^{\frac{3}{2}} : \frac{2}{9} (5 x^{3} + 3 x - 2)^{\frac{3}{2}}

= \frac{2}{9} (5 x^{3} + 3 x - 2)^{\frac{3}{2}}

解答补常数

= \frac{2}{9} (5 x^{3} + 3 x - 2)^{\frac{3}{2}} + C

\frac{d}{d x} (sinh^{2} (x))

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

\int \frac{y}{3 y + 1} d y

\frac{d y}{d x} = 3 cos (x) y, y (\frac{π}{2}) = - 2

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