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

解答

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

2 ln ∣ x - 1 ∣ - ln x^{2} + 1 - arctan (x) + C

求解步骤

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

乘开

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

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

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

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

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

化简

\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)) + 3 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x))) : 2 ln ∣ x - 1 ∣ - ln x^{2} + 1 - arctan (x)

= 2 ln ∣ x - 1 ∣ - ln x^{2} + 1 - arctan (x)

解答补常数

= 2 ln ∣ x - 1 ∣ - ln x^{2} + 1 - arctan (x) + C