integral of (4x+7)/((8x^2+28x)^3)

解答

\int \frac{4 x + 7}{( 8 x ^{2} + 28 x ) ^{3}} d x

\frac{1}{10976 x} - \frac{1}{6272 x ^{2}} - \frac{1}{5488 ( 2 x + 7 )} - \frac{1}{1568 ( 2 x + 7 ) ^{2}} + C

求解步骤

\int \frac{4 x + 7}{( 8 x ^{2} + 28 x ) ^{3}} d x

将

\frac{4 x + 7}{( 8 x ^{2} + 28 x ) ^{3}}

用部份分式展开:

- \frac{1}{10976 x ^{2}} + \frac{1}{3136 x ^{3}} + \frac{1}{2744 ( 2 x + 7 ) ^{2}} + \frac{1}{392 ( 2 x + 7 ) ^{3}}

= \int - \frac{1}{10976 x ^{2}} + \frac{1}{3136 x ^{3}} + \frac{1}{2744 ( 2 x + 7 ) ^{2}} + \frac{1}{392 ( 2 x + 7 ) ^{3}} 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{1}{10976 x ^{2}} d x + \int \frac{1}{3136 x ^{3}} d x + \int \frac{1}{2744 ( 2 x + 7 ) ^{2}} d x + \int \frac{1}{392 ( 2 x + 7 ) ^{3}} d x

\int \frac{1}{10976 x ^{2}} d x = - \frac{1}{10976 x}

\int \frac{1}{3136 x ^{3}} d x = - \frac{1}{6272 x ^{2}}

\int \frac{1}{2744 ( 2 x + 7 ) ^{2}} d x = - \frac{1}{5488 ( 2 x + 7 )}

\int \frac{1}{392 ( 2 x + 7 ) ^{3}} d x = - \frac{1}{1568 ( 2 x + 7 ) ^{2}}

= - (- \frac{1}{10976 x}) - \frac{1}{6272 x ^{2}} - \frac{1}{5488 ( 2 x + 7 )} - \frac{1}{1568 ( 2 x + 7 ) ^{2}}

化简

= \frac{1}{10976 x} - \frac{1}{6272 x ^{2}} - \frac{1}{5488 ( 2 x + 7 )} - \frac{1}{1568 ( 2 x + 7 ) ^{2}}

解答补常数

= \frac{1}{10976 x} - \frac{1}{6272 x ^{2}} - \frac{1}{5488 ( 2 x + 7 )} - \frac{1}{1568 ( 2 x + 7 ) ^{2}} + C

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