integral of (2x+4)/((3x^2+12x)^5)

解答

\int \frac{2 x + 4}{( 3 x ^{2} + 12 x ) ^{5}} d x

\frac{5}{3981312 x} - \frac{5}{1990656 x ^{2}} + \frac{1}{248832 x ^{3}} - \frac{1}{248832 x ^{4}} - \frac{5}{3981312 ( x + 4 )} - \frac{5}{1990656 ( x + 4 ) ^{2}} - \frac{1}{248832 ( x + 4 ) ^{3}} - \frac{1}{248832 ( x + 4 ) ^{4}} + C

求解步骤

\int \frac{2 x + 4}{( 3 x ^{2} + 12 x ) ^{5}} d x

将

\frac{2 x + 4}{( 3 x ^{2} + 12 x ) ^{5}}

用部份分式展开:

- \frac{5}{3981312 x ^{2}} + \frac{5}{995328 x ^{3}} - \frac{1}{82944 x ^{4}} + \frac{1}{62208 x ^{5}} + \frac{5}{3981312 ( x + 4 ) ^{2}} + \frac{5}{995328 ( x + 4 ) ^{3}} + \frac{1}{82944 ( x + 4 ) ^{4}} + \frac{1}{62208 ( x + 4 ) ^{5}}

= \int - \frac{5}{3981312 x ^{2}} + \frac{5}{995328 x ^{3}} - \frac{1}{82944 x ^{4}} + \frac{1}{62208 x ^{5}} + \frac{5}{3981312 ( x + 4 ) ^{2}} + \frac{5}{995328 ( x + 4 ) ^{3}} + \frac{1}{82944 ( x + 4 ) ^{4}} + \frac{1}{62208 ( x + 4 ) ^{5}} d x

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{5}{3981312 x ^{2}} d x + \int \frac{5}{995328 x ^{3}} d x - \int \frac{1}{82944 x ^{4}} d x + \int \frac{1}{62208 x ^{5}} d x + \int \frac{5}{3981312 ( x + 4 ) ^{2}} d x + \int \frac{5}{995328 ( x + 4 ) ^{3}} d x + \int \frac{1}{82944 ( x + 4 ) ^{4}} d x + \int \frac{1}{62208 ( x + 4 ) ^{5}} d x

\int \frac{5}{3981312 x ^{2}} d x = - \frac{5}{3981312 x}

\int \frac{5}{995328 x ^{3}} d x = - \frac{5}{1990656 x ^{2}}

\int \frac{1}{82944 x ^{4}} d x = - \frac{1}{248832 x ^{3}}

\int \frac{1}{62208 x ^{5}} d x = - \frac{1}{248832 x ^{4}}

\int \frac{5}{3981312 ( x + 4 ) ^{2}} d x = - \frac{5}{3981312 ( x + 4 )}

\int \frac{5}{995328 ( x + 4 ) ^{3}} d x = - \frac{5}{1990656 ( x + 4 ) ^{2}}

\int \frac{1}{82944 ( x + 4 ) ^{4}} d x = - \frac{1}{248832 ( x + 4 ) ^{3}}

\int \frac{1}{62208 ( x + 4 ) ^{5}} d x = - \frac{1}{248832 ( x + 4 ) ^{4}}

= - (- \frac{5}{3981312 x}) - \frac{5}{1990656 x ^{2}} - (- \frac{1}{248832 x ^{3}}) - \frac{1}{248832 x ^{4}} - \frac{5}{3981312 ( x + 4 )} - \frac{5}{1990656 ( x + 4 ) ^{2}} - \frac{1}{248832 ( x + 4 ) ^{3}} - \frac{1}{248832 ( x + 4 ) ^{4}}

化简

= \frac{5}{3981312 x} - \frac{5}{1990656 x ^{2}} + \frac{1}{248832 x ^{3}} - \frac{1}{248832 x ^{4}} - \frac{5}{3981312 ( x + 4 )} - \frac{5}{1990656 ( x + 4 ) ^{2}} - \frac{1}{248832 ( x + 4 ) ^{3}} - \frac{1}{248832 ( x + 4 ) ^{4}}

解答补常数

= \frac{5}{3981312 x} - \frac{5}{1990656 x ^{2}} + \frac{1}{248832 x ^{3}} - \frac{1}{248832 x ^{4}} - \frac{5}{3981312 ( x + 4 )} - \frac{5}{1990656 ( x + 4 ) ^{2}} - \frac{1}{248832 ( x + 4 ) ^{3}} - \frac{1}{248832 ( x + 4 ) ^{4}} + C

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