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

解答

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

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

求解步骤

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

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

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

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

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

解答补常数

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