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受欢迎的微积分 >

integral of (4x)/(sqrt(1-x^4))

解答

\int \frac{4 x}{1 - x ^{4}} d x

解答

2 arcsin (x^{2}) + C

求解步骤

\int \frac{4 x}{1 - 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}{1 - x ^{4}} d x

使用换元积分法

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

使用积分常见定则:

\int \frac{1}{1 - u ^{2}} d u = arcsin (u)

= 4 \cdot \frac{1}{2} arcsin (u)

u = x^{2}

代回

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

化简

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

= 2 arcsin (x^{2})

解答补常数

= 2 arcsin (x^{2}) + C

作图

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