integral of x*ln^3(x)

解答

\int x \cdot ln^{3} (x) d x

\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}) + C

求解步骤

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

使用分布积分法

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

\int \frac{3}{2} x ln^{2} (x) d x = \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})

= \frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})

解答补常数

= \frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}) + C

x \to 2 lim (2 x^{2} - 3 x + 5)

in v erse l a pl a ce \frac{s ^{2}}{( s ^{2} + \frac{1}{4} )}

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

t an g e n t sin (3 x)

d er i v a t i v e y = e^{x} + 1