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受欢迎的微积分 >

integral of ((sqrt(x)+1)^4)/(2sqrt(x))

解答

\int \frac{( x + 1 ) ^{4}}{2 x} d x

解答

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

求解步骤

\int \frac{( x + 1 ) ^{4}}{2 x} d x

提出常数:

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

= \frac{1}{2} \cdot \int \frac{( x + 1 ) ^{4}}{x} d x

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = x + 1

代回

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

化简

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

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

解答补常数

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

作图

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