integral of x(x^2+3)^4

解答

\int x (x^{2} + 3)^{4} d x

\frac{1}{10} (x^{2} + 3)^{5} + C

求解步骤

\int x (x^{2} + 3)^{4} d x

使用换元积分法

= \int \frac{u ^{4}}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int u^{4} d u

使用幂法则

= \frac{1}{2} \cdot \frac{u ^{5}}{5}

u = x^{2} + 3

代回

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

化简

\frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{5}}{5} : \frac{1}{10} (x^{2} + 3)^{5}

= \frac{1}{10} (x^{2} + 3)^{5}

解答补常数

= \frac{1}{10} (x^{2} + 3)^{5} + C

x \to 0 lim (8 x)

\frac{\partial}{\partial x} (3 x^{2} - 5 y)

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