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/(1-6x-3x^2)

解答

\int \frac{1}{1 - 6 x - 3 x ^{2}} d x

解答

\frac{1}{4 3} (ln \frac{3}{2} (x + 1) + 1 - ln \frac{3}{2} (x + 1) - 1) + C

求解步骤

\int \frac{1}{1 - 6 x - 3 x ^{2}} d x

因式分解

1 - 6 x - 3 x^{2} : - (- 1 + 6 x + 3 x^{2})

= \int \frac{1}{- ( - 1 + 6 x + 3 x ^{2} )} d x

提出常数:

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

= - \int \frac{1}{- 1 + 6 x + 3 x ^{2}} d x

对

- 1 + 6 x + 3 x^{2}

配方:

3 (x + 1)^{2} - 4

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

使用换元积分法

= - \int \frac{1}{3 u ^{2} - 4} d u

使用换元积分法

= - \int \frac{1}{2 3 ( v ^{2} - 1 )} d v

提出常数:

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

= - \frac{1}{2 3} \cdot \int \frac{1}{v ^{2} - 1} d v

因式分解

v^{2} - 1 : - (- v^{2} + 1)

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= - \frac{1}{2 3} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

代回

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

化简

- \frac{1}{2 3} - \frac{ln \frac{3}{2} ( x + 1 ) + 1}{2} - \frac{ln \frac{3}{2} ( x + 1 ) - 1}{2} : \frac{1}{4 3} (ln \frac{3}{2} (x + 1) + 1 - ln \frac{3}{2} (x + 1) - 1)

= \frac{1}{4 3} (ln \frac{3}{2} (x + 1) + 1 - ln \frac{3}{2} (x + 1) - 1)

解答补常数

= \frac{1}{4 3} (ln \frac{3}{2} (x + 1) + 1 - ln \frac{3}{2} (x + 1) - 1) + C

作图

流行的例子

integral of sqrt(x)sqrt(9-x)

\int x 9 - x d x

integral of (e^t)/(1-e^{2t)}

\int \frac{e ^{t}}{1 - e ^{2 t}} d t

integral of (sin(x))/(3+cos(2x))

\int \frac{sin ( x )}{3 + cos ( 2 x )} d x

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

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

integral of cos(sqrt(u))

\int cos (u) d u