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

解

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

(28 + 28 x) (\frac{1}{4} ln ∣ s - 1 ∣ + \frac{1}{2 ( s - 1 )} - \frac{1}{4 ( s - 1 ) ^{2}} + \frac{1}{4} (- \frac{1}{2} ln s^{2} + 1 + arctan (s))) + C

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

\int \frac{1}{( - 1 + s ) ^{3} ( 1 + s ^{2} )} d s = \frac{1}{4} ln ∣ s - 1 ∣ + \frac{1}{2 ( s - 1 )} - \frac{1}{4 ( s - 1 ) ^{2}} + \frac{1}{4} (- \frac{1}{2} ln s^{2} + 1 + arctan (s))

= (28 + 28 x) (\frac{1}{4} ln ∣ s - 1 ∣ + \frac{1}{2 ( s - 1 )} - \frac{1}{4 ( s - 1 ) ^{2}} + \frac{1}{4} (- \frac{1}{2} ln s^{2} + 1 + arctan (s)))

定数を解答に追加する

= (28 + 28 x) (\frac{1}{4} ln ∣ s - 1 ∣ + \frac{1}{2 ( s - 1 )} - \frac{1}{4 ( s - 1 ) ^{2}} + \frac{1}{4} (- \frac{1}{2} ln s^{2} + 1 + arctan (s))) + C