integral of (12)/(x^2-1)

解答

\int \frac{12}{x ^{2} - 1} d x

- 12 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

求解步骤

\int \frac{12}{x ^{2} - 1} d x

提出常数:

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

= 12 \cdot \int \frac{1}{x ^{2} - 1} d x

因式分解

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

= 12 \cdot \int \frac{1}{- ( - x ^{2} + 1 )} d x

提出常数:

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

= 12 (- \int \frac{1}{- x ^{2} + 1} d x)

使用积分常见定则:

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

= 12 (- (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}))

化简

12 (- (\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2})) : - 12 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

= - 12 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

解答补常数

= - 12 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

\frac{d}{d x} (\frac{6 x}{e ^{x}})

\int x^{2} cos (w x) d x

f (I) = \frac{110 I}{I ^{2} + I + 1}

d er i v a t i v e arctan (4 t)

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