integral of (24s+24)/((s^2+1)(s-1)^3)

解答

\int \frac{24 s + 24}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

12 arctan (s) + \frac{12}{s - 1} - \frac{12}{( s - 1 ) ^{2}} + C

求解步骤

\int \frac{24 s + 24}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

将

\frac{24 s + 24}{( s ^{2} + 1 ) ( s - 1 ) ^{3}}

用部份分式展开:

\frac{12}{s ^{2} + 1} - \frac{12}{( s - 1 ) ^{2}} + \frac{24}{( s - 1 ) ^{3}}

= \int \frac{12}{s ^{2} + 1} - \frac{12}{( s - 1 ) ^{2}} + \frac{24}{( 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{12}{s ^{2} + 1} d s - \int \frac{12}{( s - 1 ) ^{2}} d s + \int \frac{24}{( s - 1 ) ^{3}} d s

\int \frac{12}{s ^{2} + 1} d s = 12 arctan (s)

\int \frac{12}{( s - 1 ) ^{2}} d s = - \frac{12}{s - 1}

\int \frac{24}{( s - 1 ) ^{3}} d s = - \frac{12}{( s - 1 ) ^{2}}

= 12 arctan (s) - (- \frac{12}{s - 1}) - \frac{12}{( s - 1 ) ^{2}}

化简

= 12 arctan (s) + \frac{12}{s - 1} - \frac{12}{( s - 1 ) ^{2}}

解答补常数

= 12 arctan (s) + \frac{12}{s - 1} - \frac{12}{( s - 1 ) ^{2}} + C

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