zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of 1/(sqrt(1-6x-9x^2))

解答

\int \frac{1}{1 - 6 x - 9 x ^{2}} d x

解答

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

求解步骤

\int \frac{1}{1 - 6 x - 9 x ^{2}} d x

对

1 - 6 x - 9 x^{2}

配方:

- 9 (x + \frac{1}{3})^{2} + 2

= \int \frac{1}{- 9 ( x + \frac{1}{3} ) ^{2} + 2} d x

使用换元积分法

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

使用三角换元法

= \int \frac{1}{3} d v

提出常数:

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

= \frac{1}{3} \cdot \int 1 d v

常数积分:

\int a d x = a x

= \frac{1}{3} \cdot 1 \cdot v

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

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

\int x^{2} (1 - x^{3}) d x

integral of sinh^3(2x)cosh(2x)

\int sinh^{3} (2 x) cosh (2 x) d x

integral of (e^x+2)/(e^{2x)+2e^x+5}

\int \frac{e ^{x} + 2}{e ^{2 x} + 2 e ^{x} + 5} d x

integral of (18t-5)/(sqrt(9t^2-5t+2))

\int \frac{18 t - 5}{9 t ^{2} - 5 t + 2} d t

integral of sin^{1/3}(x)cos(x)

\int sin^{\frac{1}{3}} (x) cos (x) d x