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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

将

\frac{1}{( u - 1 ) u ^{2}}

用部份分式展开:

\frac{1}{u - 1} - \frac{1}{u} - \frac{1}{u ^{2}}

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

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{1}{u - 1} d u - \int \frac{1}{u} d u - \int \frac{1}{u ^{2}} d u

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u - 1 ∣ - ln ∣ u ∣ - (- \frac{1}{u})

u = 1 + x

代回

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

化简

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

解答补常数

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

作图

流行的例子

integral of sqrt((3+x)/(1-x))

\int \frac{3 + x}{1 - x} d x

integral of e^{ikx}

\int e^{ik x} d x

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

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

integral of (sin(2x))/(sqrt(3-cos^4(x)))

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

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

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