integral of (28s+28)/((s^2+1)(s-1)^3)

解答

\int \frac{28 s + 28}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

14 arctan (s) + \frac{14}{s - 1} - \frac{14}{( s - 1 ) ^{2}} + C

求解步骤

\int \frac{28 s + 28}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

将

\frac{28 s + 28}{( s ^{2} + 1 ) ( s - 1 ) ^{3}}

用部份分式展开:

\frac{14}{s ^{2} + 1} - \frac{14}{( s - 1 ) ^{2}} + \frac{28}{( s - 1 ) ^{3}}

= \int \frac{14}{s ^{2} + 1} - \frac{14}{( s - 1 ) ^{2}} + \frac{28}{( s - 1 ) ^{3}} d s

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 \frac{14}{s ^{2} + 1} d s - \int \frac{14}{( s - 1 ) ^{2}} d s + \int \frac{28}{( s - 1 ) ^{3}} d s

\int \frac{14}{s ^{2} + 1} d s = 14 arctan (s)

\int \frac{14}{( s - 1 ) ^{2}} d s = - \frac{14}{s - 1}

\int \frac{28}{( s - 1 ) ^{3}} d s = - \frac{14}{( s - 1 ) ^{2}}

= 14 arctan (s) - (- \frac{14}{s - 1}) - \frac{14}{( s - 1 ) ^{2}}

化简

= 14 arctan (s) + \frac{14}{s - 1} - \frac{14}{( s - 1 ) ^{2}}

解答补常数

= 14 arctan (s) + \frac{14}{s - 1} - \frac{14}{( s - 1 ) ^{2}} + C

\frac{\partial}{\partial x} (3 x + 4 y^{2} + 3 x^{2} y^{3})

\int x - 2 π d x

\int (x) \frac{3}{4} x^{2} - 4 x + 8 d x

\frac{d}{d x} ((2 x^{3}) (4 x^{2}))

s l o p e in t erce pt (0, 5), (- 1.5, 1)