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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

1 - x - x^{2}

配方:

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

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

使用换元积分法

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

化简

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

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

解答补常数

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

作图

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\int (8 + x^{2})^{- \frac{3}{2}} d x

integral of 1/(15x-2)

\int \frac{1}{15 x - 2} d x

integral of-2cos^3(x)(cos^2(x))''sin(x)

\int - 2 cos^{3} (x) (cos^{2} (x))^{''} sin (x) d x

integral of (24x)/(2x+3)

\int \frac{24 x}{2 x + 3} d x

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\int \frac{5 x + 3}{x ^{2} - 4} d x