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受欢迎的微积分 >

integral of (4x)/(sqrt(9-x^4))

解答

\int \frac{4 x}{9 - x ^{4}} d x

解答

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

求解步骤

\int \frac{4 x}{9 - x ^{4}} d x

提出常数:

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

= 4 \cdot \int \frac{x}{9 - x ^{4}} d x

使用换元积分法

= 4 \cdot \int \frac{1}{2 9 - u ^{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 \frac{1}{9 - u ^{2}} d u

使用三角换元法

= 4 \cdot \frac{1}{2} \cdot \int 1 d v

常数积分:

\int a d x = a x

= 4 \cdot \frac{1}{2} \cdot 1 \cdot v

代回

= 4 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{1}{3} x^{2})

化简

4 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{1}{3} x^{2}) : 2 arcsin (\frac{1}{3} x^{2})

= 2 arcsin (\frac{1}{3} x^{2})

解答补常数

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

作图

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