integral of (xln(x))/((1-x^2)^{1/2)}

解答

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

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

求解步骤

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

使用分布积分法

= x ln (x) arcsin (x) - \int (ln (x) + 1) arcsin (x) d x

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

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

化简

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

解答补常数

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

in v erse l a pl a ce \frac{1}{( s ^{2} + 0.5 s + 1 )}

y^{^{''}} - 6 y^{^{'}} + 9 y = \frac{e ^{3 x}}{x ^{2}}

a re a x = 15 - y^{2}, x = y^{2} - 3

\int_{0}^{\frac{π}{8}} arctan (2 x) d x

\frac{\partial}{\partial w} (w arctan (uv))