integral of x*ln(5+x)

解答

\int x \cdot ln (5 + x) d x

ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

求解步骤

\int x ln (5 + x) d x

使用换元积分法

= \int (u - 5) ln (u) d u

使用分布积分法

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u

\int \frac{1}{2} (u - 10) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

u = 5 + x

代回

= ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x))

化简

ln (5 + x) (\frac{( 5 + x ) ^{2}}{2} - 5 (5 + x)) - \frac{1}{2} (\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x)) : ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75)

解答补常数

= ln (5 + x) (\frac{1}{2} (5 + x)^{2} - 5 (5 + x)) - \frac{1}{4} (x^{2} - 10 x - 75) + C

\int \frac{π}{2} - 5 x^{- 0.5} d x

\frac{d ^{3}}{d x ^{3}} (x^{4} - 8 x^{3})

\frac{d}{d x} (e^{c o s (x + 3)})

- 8 t^{2} y^{^{''}} - 4 t (t - 4) y^{^{'}} + 4 (t - 4) y = 0

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