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受欢迎的微积分 >

integral of (2x)/(sqrt(-5-6x^2-x^4))

解答

\int \frac{2 x}{- 5 - 6 x ^{2} - x ^{4}} d x

解答

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

求解步骤

\int \frac{2 x}{- 5 - 6 x ^{2} - x ^{4}} d x

提出常数:

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

= 2 \cdot \int \frac{x}{- 5 - 6 x ^{2} - x ^{4}} d x

使用换元积分法

= 2 \cdot \int \frac{1}{2 - 5 - 6 u - u ^{2}} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- 5 - 6 u - u ^{2}} d u

对

- 5 - 6 u - u^{2}

配方:

- (u + 3)^{2} + 4

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- ( u + 3 ) ^{2} + 4} d u

使用换元积分法

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

使用三角换元法

= 2 \cdot \frac{1}{2} \cdot \int 1 d w

常数积分:

\int a d x = a x

= 2 \cdot \frac{1}{2} \cdot 1 \cdot w

代回

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

化简

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

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

解答补常数

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

作图

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