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受欢迎的微积分 >

integral of 4x*[x^2+5]^3

解答

\int 4 x \cdot [x^{2} + 5]^{3} d x

解答

\frac{1}{2} (x^{2} + 5)^{4} + C

求解步骤

\int 4 x [x^{2} + 5]^{3} d x

提出常数:

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

= 4 \cdot \int x [x^{2} + 5]^{3} d x

化简

(x^{2} + 5)^{3} : (x^{2} + 5)^{2} (x^{2} + 5)

= 4 \cdot \int x (x^{2} + 5)^{2} (x^{2} + 5) d x

化简

x (x^{2} + 5)^{2} (x^{2} + 5) : x (x^{2} + 5)^{3}

= 4 \cdot \int x (x^{2} + 5)^{3} d x

使用换元积分法

= 4 \cdot \int \frac{u ^{3}}{2} d u

提出常数:

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

= 4 \cdot \frac{1}{2} \cdot \int u^{3} d u

使用幂法则

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

u = x^{2} + 5

代回

= 4 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 5 ) ^{4}}{4}

化简

4 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 5 ) ^{4}}{4} : \frac{1}{2} (x^{2} + 5)^{4}

= \frac{1}{2} (x^{2} + 5)^{4}

解答补常数

= \frac{1}{2} (x^{2} + 5)^{4} + C

作图

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