integral of x^3ln(x^2+3)

解答

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

\frac{1}{4} x^{4} ln (x^{2} + 3) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{3 x ^{2}}{2} + \frac{9}{2} ln x^{2} + 3) + C

求解步骤

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

使用分布积分法

= \frac{1}{4} x^{4} ln (x^{2} + 3) - \int \frac{x ^{5}}{2 ( x ^{2} + 3 )} d x

\int \frac{x ^{5}}{2 ( x ^{2} + 3 )} d x = \frac{1}{2} (\frac{x ^{4}}{4} - \frac{3 x ^{2}}{2} + \frac{9}{2} ln x^{2} + 3)

= \frac{1}{4} x^{4} ln (x^{2} + 3) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{3 x ^{2}}{2} + \frac{9}{2} ln x^{2} + 3)

解答补常数

= \frac{1}{4} x^{4} ln (x^{2} + 3) - \frac{1}{2} (\frac{x ^{4}}{4} - \frac{3 x ^{2}}{2} + \frac{9}{2} ln x^{2} + 3) + C

\int - cos (- x) d x

\int 8 (9 + x)^{5} d x

\int \frac{4 x ^{4} - 4 x ^{3} + 7 x}{x ^{2}} d x

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

\int sin (x) 7 + cos (x) d x