integral of (2x+3)/((2x^2+6x)^4)

解答

\int \frac{2 x + 3}{( 2 x ^{2} + 6 x ) ^{4}} d x

- \frac{1}{1944 x} + \frac{1}{1296 x ^{2}} - \frac{1}{1296 x ^{3}} + \frac{1}{1944 ( x + 3 )} + \frac{1}{1296 ( x + 3 ) ^{2}} + \frac{1}{1296 ( x + 3 ) ^{3}} + C

求解步骤

\int \frac{2 x + 3}{( 2 x ^{2} + 6 x ) ^{4}} d x

将

\frac{2 x + 3}{( 2 x ^{2} + 6 x ) ^{4}}

用部份分式展开:

\frac{1}{1944 x ^{2}} - \frac{1}{648 x ^{3}} + \frac{1}{432 x ^{4}} - \frac{1}{1944 ( x + 3 ) ^{2}} - \frac{1}{648 ( x + 3 ) ^{3}} - \frac{1}{432 ( x + 3 ) ^{4}}

= \int \frac{1}{1944 x ^{2}} - \frac{1}{648 x ^{3}} + \frac{1}{432 x ^{4}} - \frac{1}{1944 ( x + 3 ) ^{2}} - \frac{1}{648 ( x + 3 ) ^{3}} - \frac{1}{432 ( x + 3 ) ^{4}} 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}{1944 x ^{2}} d x - \int \frac{1}{648 x ^{3}} d x + \int \frac{1}{432 x ^{4}} d x - \int \frac{1}{1944 ( x + 3 ) ^{2}} d x - \int \frac{1}{648 ( x + 3 ) ^{3}} d x - \int \frac{1}{432 ( x + 3 ) ^{4}} d x

\int \frac{1}{1944 x ^{2}} d x = - \frac{1}{1944 x}

\int \frac{1}{648 x ^{3}} d x = - \frac{1}{1296 x ^{2}}

\int \frac{1}{432 x ^{4}} d x = - \frac{1}{1296 x ^{3}}

\int \frac{1}{1944 ( x + 3 ) ^{2}} d x = - \frac{1}{1944 ( x + 3 )}

\int \frac{1}{648 ( x + 3 ) ^{3}} d x = - \frac{1}{1296 ( x + 3 ) ^{2}}

\int \frac{1}{432 ( x + 3 ) ^{4}} d x = - \frac{1}{1296 ( x + 3 ) ^{3}}

= - \frac{1}{1944 x} - (- \frac{1}{1296 x ^{2}}) - \frac{1}{1296 x ^{3}} - (- \frac{1}{1944 ( x + 3 )}) - (- \frac{1}{1296 ( x + 3 ) ^{2}}) - (- \frac{1}{1296 ( x + 3 ) ^{3}})

化简

= - \frac{1}{1944 x} + \frac{1}{1296 x ^{2}} - \frac{1}{1296 x ^{3}} + \frac{1}{1944 ( x + 3 )} + \frac{1}{1296 ( x + 3 ) ^{2}} + \frac{1}{1296 ( x + 3 ) ^{3}}

解答补常数

= - \frac{1}{1944 x} + \frac{1}{1296 x ^{2}} - \frac{1}{1296 x ^{3}} + \frac{1}{1944 ( x + 3 )} + \frac{1}{1296 ( x + 3 ) ^{2}} + \frac{1}{1296 ( x + 3 ) ^{3}} + C

\frac{d x}{d t} = \frac{1}{t ( 1 - t )}

\frac{d}{d t} (x y)

\int (x - \frac{1}{x ^{2}}) d x

\int ln (5^{x}) d x

d er i v a t i v e 9 t^{2} + t^{3}