integral of (4x+2)/((x+1)(x^2+1))

解答

\int \frac{4 x + 2}{( x + 1 ) ( x ^{2} + 1 )} d x

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

求解步骤

\int \frac{4 x + 2}{( x + 1 ) ( x ^{2} + 1 )} d x

乘开

\frac{4 x + 2}{( x + 1 ) ( x ^{2} + 1 )} : \frac{4 x}{( x + 1 ) ( x ^{2} + 1 )} + \frac{2}{( 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{4 x}{( x + 1 ) ( x ^{2} + 1 )} d x + \int \frac{2}{( x + 1 ) ( x ^{2} + 1 )} d x

\int \frac{4 x}{( x + 1 ) ( x ^{2} + 1 )} d x = 4 (- \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x)))

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

= 4 (- \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x))) + ln ∣ x + 1 ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x)

化简

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

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

解答补常数

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

\frac{d}{d x} (1000 (1 - \frac{x}{500})^{2})

\frac{d y}{d x} = 4 + 9 x + 32 y + 72 x y

\int_{0}^{t} e^{s} sin (t - s) d s

d er i v a t i v e f (s) = - \frac{12}{s ^{5}}

\frac{d}{d x} (x ln (x^{15}))