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受欢迎的微积分 >

integral of 1/(sqrt(2-3x-x^2))

解答

\int \frac{1}{2 - 3 x - x ^{2}} d x

解答

arcsin (\frac{1}{17} (2 x + 3)) + C

求解步骤

\int \frac{1}{2 - 3 x - x ^{2}} d x

对

2 - 3 x - x^{2}

配方:

- (x + \frac{3}{2})^{2} + \frac{17}{4}

= \int \frac{1}{- ( x + \frac{3}{2} ) ^{2} + \frac{17}{4}} d x

使用换元积分法

= \int \frac{2}{- 4 u ^{2} + 17} d u

提出常数:

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

= 2 \cdot \int \frac{1}{- 4 u ^{2} + 17} d u

使用三角换元法

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

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 1 d v

常数积分:

\int a d x = a x

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

代回

= 2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{17} (x + \frac{3}{2}))

化简

2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{17} (x + \frac{3}{2})) : arcsin (\frac{1}{17} (2 x + 3))

= arcsin (\frac{1}{17} (2 x + 3))

解答补常数

= arcsin (\frac{1}{17} (2 x + 3)) + C

作图

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