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受欢迎的微积分 >

积分 x/(((x-1)(x-2)(x-3)))

解答

积分

\frac{x}{( ( x - 1 ) ( x - 2 ) ( x - 3 ) )}

解答

\frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣ + C

求解步骤

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

使用换元积分法

= \int \frac{u + 2}{u ( u + 1 ) ( u - 1 )} d u

乘开

\frac{u + 2}{u ( u + 1 ) ( u - 1 )} : \frac{u}{u ( u + 1 ) ( u - 1 )} + \frac{2}{u ( u + 1 ) ( u - 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

= \int \frac{u}{u ( u + 1 ) ( u - 1 )} d u + \int \frac{2}{u ( u + 1 ) ( u - 1 )} d u

\int \frac{u}{u ( u + 1 ) ( u - 1 )} d u = - \frac{1}{2} ln ∣ u + 1 ∣ + \frac{1}{2} ln ∣ u - 1 ∣

\int \frac{2}{u ( u + 1 ) ( u - 1 )} d u = - 2 ln ∣ u ∣ + ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣

= - \frac{1}{2} ln ∣ u + 1 ∣ + \frac{1}{2} ln ∣ u - 1 ∣ - 2 ln ∣ u ∣ + ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣

u = x - 2

代回

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

化简

- \frac{1}{2} ln ∣ x - 2 + 1 ∣ + \frac{1}{2} ln ∣ x - 2 - 1 ∣ - 2 ln ∣ x - 2 ∣ + ln ∣ x - 2 + 1 ∣ + ln ∣ x - 2 - 1 ∣ : \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣

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

解答补常数

= \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣ + C

作图

流行的例子

积分 (a-x^2)^{1/2}

in t e g r a l (a - x^{2})^{\frac{1}{2}}

积分 sin^3(x)cos^3(x)

in t e g r a l sin^{3} (x) cos^{3} (x)

f(x)=(((x-1))/((x+1)))^{1/2}

f (x) = (\frac{( x - 1 )}{( x + 1 )})^{\frac{1}{2}}

积分 sin^2(x)cos^3(x)

in t e g r a l sin^{2} (x) cos^{3} (x)

in t e g r a l \frac{1}{v ^{4}}