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 (8x-5)/(3x^2-5x+2)

解答

\int \frac{8 x - 5}{3 x ^{2} - 5 x + 2} d x

解答

\frac{4}{3} ln 36 x^{2} - 60 x + 24 - \frac{20}{3} (ln ∣ 6 x - 4 ∣ - ln ∣ 6 x - 6 ∣) + 5 ln ∣ 6 x - 4 ∣ - 5 ln ∣ 6 x - 6 ∣ + C

求解步骤

\int \frac{8 x - 5}{3 x ^{2} - 5 x + 2} d x

对

3 x^{2} - 5 x + 2

配方:

3 (x - \frac{5}{6})^{2} - \frac{1}{12}

= \int \frac{8 x - 5}{3 ( x - \frac{5}{6} ) ^{2} - \frac{1}{12}} d x

使用换元积分法

= \int \frac{12 ( 8 ( u + \frac{5}{6} ) - 5 )}{36 u ^{2} - 1} d u

提出常数:

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

= 12 \cdot \int \frac{8 ( u + \frac{5}{6} ) - 5}{36 u ^{2} - 1} d u

乘开

\frac{8 ( u + \frac{5}{6} ) - 5}{36 u ^{2} - 1} : \frac{8 ( u + \frac{5}{6} )}{36 u ^{2} - 1} - \frac{5}{36 u ^{2} - 1}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 12 (\int \frac{8 ( u + \frac{5}{6} )}{36 u ^{2} - 1} d u - \int \frac{5}{36 u ^{2} - 1} d u)

\int \frac{8 ( u + \frac{5}{6} )}{36 u ^{2} - 1} d u = 8 (\frac{1}{72} ln 36 u^{2} - 1 - \frac{5}{72} (ln ∣ 6 u + 1 ∣ - ln ∣ 6 u - 1 ∣))

\int \frac{5}{36 u ^{2} - 1} d u = - \frac{5}{12} (ln ∣ 6 u + 1 ∣ - ln ∣ 6 u - 1 ∣)

= 12 (8 (\frac{1}{72} ln 36 u^{2} - 1 - \frac{5}{72} (ln ∣ 6 u + 1 ∣ - ln ∣ 6 u - 1 ∣)) - (- \frac{5}{12} (ln ∣ 6 u + 1 ∣ - ln ∣ 6 u - 1 ∣)))

u = x - \frac{5}{6}

代回

= 12 (8 (\frac{1}{72} ln 36 (x - \frac{5}{6})^{2} - 1 - \frac{5}{72} (ln 6 (x - \frac{5}{6}) + 1 - ln 6 (x - \frac{5}{6}) - 1)) - (- \frac{5}{12} (ln 6 (x - \frac{5}{6}) + 1 - ln 6 (x - \frac{5}{6}) - 1)))

化简

12 (8 (\frac{1}{72} ln 36 (x - \frac{5}{6})^{2} - 1 - \frac{5}{72} (ln 6 (x - \frac{5}{6}) + 1 - ln 6 (x - \frac{5}{6}) - 1)) - (- \frac{5}{12} (ln 6 (x - \frac{5}{6}) + 1 - ln 6 (x - \frac{5}{6}) - 1))) : \frac{4}{3} ln 36 x^{2} - 60 x + 24 - \frac{20}{3} (ln ∣ 6 x - 4 ∣ - ln ∣ 6 x - 6 ∣) + 5 ln ∣ 6 x - 4 ∣ - 5 ln ∣ 6 x - 6 ∣

= \frac{4}{3} ln 36 x^{2} - 60 x + 24 - \frac{20}{3} (ln ∣ 6 x - 4 ∣ - ln ∣ 6 x - 6 ∣) + 5 ln ∣ 6 x - 4 ∣ - 5 ln ∣ 6 x - 6 ∣

解答补常数

= \frac{4}{3} ln 36 x^{2} - 60 x + 24 - \frac{20}{3} (ln ∣ 6 x - 4 ∣ - ln ∣ 6 x - 6 ∣) + 5 ln ∣ 6 x - 4 ∣ - 5 ln ∣ 6 x - 6 ∣ + C

作图

流行的例子

integral of (3a)/(4t)

\int \frac{3 a}{4 t} d t

integral of 1/(sqrt(2x^2+5x+3))

\int \frac{1}{2 x ^{2} + 5 x + 3} d x

integral of 1 (T)

\int \frac{d}{1} (T) d x

in t e g r a l \frac{1}{u ^{3}}

integral of ((4x^4+3x^3-2x)/x)

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