integral of (3x+2)/((6x^2+8x)^5)

解答

\int \frac{3 x + 2}{( 6 x ^{2} + 8 x ) ^{5}} d x

\frac{135}{1048576 x} - \frac{45}{524288 x ^{2}} + \frac{3}{65536 x ^{3}} - \frac{1}{65536 x ^{4}} - \frac{405}{1048576 ( 3 x + 4 )} - \frac{405}{524288 ( 3 x + 4 ) ^{2}} - \frac{81}{65536 ( 3 x + 4 ) ^{3}} - \frac{81}{65536 ( 3 x + 4 ) ^{4}} + C

求解步骤

\int \frac{3 x + 2}{( 6 x ^{2} + 8 x ) ^{5}} d x

将

\frac{3 x + 2}{( 6 x ^{2} + 8 x ) ^{5}}

用部份分式展开:

- \frac{135}{1048576 x ^{2}} + \frac{45}{262144 x ^{3}} - \frac{9}{65536 x ^{4}} + \frac{1}{16384 x ^{5}} + \frac{1215}{1048576 ( 3 x + 4 ) ^{2}} + \frac{1215}{262144 ( 3 x + 4 ) ^{3}} + \frac{729}{65536 ( 3 x + 4 ) ^{4}} + \frac{243}{16384 ( 3 x + 4 ) ^{5}}

= \int - \frac{135}{1048576 x ^{2}} + \frac{45}{262144 x ^{3}} - \frac{9}{65536 x ^{4}} + \frac{1}{16384 x ^{5}} + \frac{1215}{1048576 ( 3 x + 4 ) ^{2}} + \frac{1215}{262144 ( 3 x + 4 ) ^{3}} + \frac{729}{65536 ( 3 x + 4 ) ^{4}} + \frac{243}{16384 ( 3 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{135}{1048576 x ^{2}} d x + \int \frac{45}{262144 x ^{3}} d x - \int \frac{9}{65536 x ^{4}} d x + \int \frac{1}{16384 x ^{5}} d x + \int \frac{1215}{1048576 ( 3 x + 4 ) ^{2}} d x + \int \frac{1215}{262144 ( 3 x + 4 ) ^{3}} d x + \int \frac{729}{65536 ( 3 x + 4 ) ^{4}} d x + \int \frac{243}{16384 ( 3 x + 4 ) ^{5}} d x

\int \frac{135}{1048576 x ^{2}} d x = - \frac{135}{1048576 x}

\int \frac{45}{262144 x ^{3}} d x = - \frac{45}{524288 x ^{2}}

\int \frac{9}{65536 x ^{4}} d x = - \frac{3}{65536 x ^{3}}

\int \frac{1}{16384 x ^{5}} d x = - \frac{1}{65536 x ^{4}}

\int \frac{1215}{1048576 ( 3 x + 4 ) ^{2}} d x = - \frac{405}{1048576 ( 3 x + 4 )}

\int \frac{1215}{262144 ( 3 x + 4 ) ^{3}} d x = - \frac{405}{524288 ( 3 x + 4 ) ^{2}}

\int \frac{729}{65536 ( 3 x + 4 ) ^{4}} d x = - \frac{81}{65536 ( 3 x + 4 ) ^{3}}

\int \frac{243}{16384 ( 3 x + 4 ) ^{5}} d x = - \frac{81}{65536 ( 3 x + 4 ) ^{4}}

= - (- \frac{135}{1048576 x}) - \frac{45}{524288 x ^{2}} - (- \frac{3}{65536 x ^{3}}) - \frac{1}{65536 x ^{4}} - \frac{405}{1048576 ( 3 x + 4 )} - \frac{405}{524288 ( 3 x + 4 ) ^{2}} - \frac{81}{65536 ( 3 x + 4 ) ^{3}} - \frac{81}{65536 ( 3 x + 4 ) ^{4}}

化简

= \frac{135}{1048576 x} - \frac{45}{524288 x ^{2}} + \frac{3}{65536 x ^{3}} - \frac{1}{65536 x ^{4}} - \frac{405}{1048576 ( 3 x + 4 )} - \frac{405}{524288 ( 3 x + 4 ) ^{2}} - \frac{81}{65536 ( 3 x + 4 ) ^{3}} - \frac{81}{65536 ( 3 x + 4 ) ^{4}}

解答补常数

= \frac{135}{1048576 x} - \frac{45}{524288 x ^{2}} + \frac{3}{65536 x ^{3}} - \frac{1}{65536 x ^{4}} - \frac{405}{1048576 ( 3 x + 4 )} - \frac{405}{524288 ( 3 x + 4 ) ^{2}} - \frac{81}{65536 ( 3 x + 4 ) ^{3}} - \frac{81}{65536 ( 3 x + 4 ) ^{4}} + C

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in t e g r a l f (x) = (x^{3} + x) (x^{2} + 2)