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受欢迎的微积分 >

integral of 4x^3(x^4+5)^6

解答

\int 4 x^{3} (x^{4} + 5)^{6} d x

解答

\frac{1}{7} (x^{4} + 5)^{7} + C

求解步骤

\int 4 x^{3} (x^{4} + 5)^{6} d x

提出常数:

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

= 4 \cdot \int x^{3} (x^{4} + 5)^{6} d x

使用换元积分法

= 4 \cdot \int \frac{u ^{6}}{4} d u

提出常数:

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

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

使用幂法则

= 4 \cdot \frac{1}{4} \cdot \frac{u ^{7}}{7}

u = x^{4} + 5

代回

= 4 \cdot \frac{1}{4} \cdot \frac{( x ^{4} + 5 ) ^{7}}{7}

化简

4 \cdot \frac{1}{4} \cdot \frac{( x ^{4} + 5 ) ^{7}}{7} : \frac{1}{7} (x^{4} + 5)^{7}

= \frac{1}{7} (x^{4} + 5)^{7}

解答补常数

= \frac{1}{7} (x^{4} + 5)^{7} + C

作图

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