integral of x^2sqrt(3+x^3)

解答

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

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

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = 3 + x^{3}

代回

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

化简

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

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

解答补常数

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

(x^{2} + y) d x + (x + e^{y}) d y = 0

d er i v a t i v e 2 t e^{- t}

s l o p e (1, - 4), (4, - 5)

\int cos (π)

x \to 3 + lim (\frac{1}{x ^{2} - 9})