integral of x^3(5+x^4)^5

解答

\int x^{3} (5 + x^{4})^{5} d x

\frac{1}{24} (5 + x^{4})^{6} + C

求解步骤

\int x^{3} (5 + x^{4})^{5} d x

使用换元积分法

= \int \frac{u ^{5}}{4} d u

提出常数:

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

= \frac{1}{4} \cdot \int u^{5} d u

使用幂法则

= \frac{1}{4} \cdot \frac{u ^{6}}{6}

u = 5 + x^{4}

代回

= \frac{1}{4} \cdot \frac{( 5 + x ^{4} ) ^{6}}{6}

化简

\frac{1}{4} \cdot \frac{( 5 + x ^{4} ) ^{6}}{6} : \frac{1}{24} (5 + x^{4})^{6}

= \frac{1}{24} (5 + x^{4})^{6}

解答补常数

= \frac{1}{24} (5 + x^{4})^{6} + C

\int_{0}^{1} (1 + 2^{x})^{2} 2^{- x} d x

\frac{\partial}{\partial x} (\cdot e)

n = 1 \sum \infty 5 \frac{32}{n ^{6}}

d er i v a t i v e f (x) = (x^{2} + 1) (2 - x)

x \to - 3 lim (2 x + 5)