integral of (6x+7)/((9x^2+21x)^4)

解答

\int \frac{6 x + 7}{( 9 x ^{2} + 21 x ) ^{4}} d x

- \frac{2}{151263 x} + \frac{1}{64827 x ^{2}} - \frac{1}{83349 x ^{3}} + \frac{2}{50421 ( 3 x + 7 )} + \frac{1}{7203 ( 3 x + 7 ) ^{2}} + \frac{1}{3087 ( 3 x + 7 ) ^{3}} + C

求解步骤

\int \frac{6 x + 7}{( 9 x ^{2} + 21 x ) ^{4}} d x

将

\frac{6 x + 7}{( 9 x ^{2} + 21 x ) ^{4}}

用部份分式展开:

\frac{2}{151263 x ^{2}} - \frac{2}{64827 x ^{3}} + \frac{1}{27783 x ^{4}} - \frac{2}{16807 ( 3 x + 7 ) ^{2}} - \frac{2}{2401 ( 3 x + 7 ) ^{3}} - \frac{1}{343 ( 3 x + 7 ) ^{4}}

= \int \frac{2}{151263 x ^{2}} - \frac{2}{64827 x ^{3}} + \frac{1}{27783 x ^{4}} - \frac{2}{16807 ( 3 x + 7 ) ^{2}} - \frac{2}{2401 ( 3 x + 7 ) ^{3}} - \frac{1}{343 ( 3 x + 7 ) ^{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{2}{151263 x ^{2}} d x - \int \frac{2}{64827 x ^{3}} d x + \int \frac{1}{27783 x ^{4}} d x - \int \frac{2}{16807 ( 3 x + 7 ) ^{2}} d x - \int \frac{2}{2401 ( 3 x + 7 ) ^{3}} d x - \int \frac{1}{343 ( 3 x + 7 ) ^{4}} d x

\int \frac{2}{151263 x ^{2}} d x = - \frac{2}{151263 x}

\int \frac{2}{64827 x ^{3}} d x = - \frac{1}{64827 x ^{2}}

\int \frac{1}{27783 x ^{4}} d x = - \frac{1}{83349 x ^{3}}

\int \frac{2}{16807 ( 3 x + 7 ) ^{2}} d x = - \frac{2}{50421 ( 3 x + 7 )}

\int \frac{2}{2401 ( 3 x + 7 ) ^{3}} d x = - \frac{1}{7203 ( 3 x + 7 ) ^{2}}

\int \frac{1}{343 ( 3 x + 7 ) ^{4}} d x = - \frac{1}{3087 ( 3 x + 7 ) ^{3}}

= - \frac{2}{151263 x} - (- \frac{1}{64827 x ^{2}}) - \frac{1}{83349 x ^{3}} - (- \frac{2}{50421 ( 3 x + 7 )}) - (- \frac{1}{7203 ( 3 x + 7 ) ^{2}}) - (- \frac{1}{3087 ( 3 x + 7 ) ^{3}})

化简

= - \frac{2}{151263 x} + \frac{1}{64827 x ^{2}} - \frac{1}{83349 x ^{3}} + \frac{2}{50421 ( 3 x + 7 )} + \frac{1}{7203 ( 3 x + 7 ) ^{2}} + \frac{1}{3087 ( 3 x + 7 ) ^{3}}

解答补常数

= - \frac{2}{151263 x} + \frac{1}{64827 x ^{2}} - \frac{1}{83349 x ^{3}} + \frac{2}{50421 ( 3 x + 7 )} + \frac{1}{7203 ( 3 x + 7 ) ^{2}} + \frac{1}{3087 ( 3 x + 7 ) ^{3}} + C

n = 1 \sum \infty \frac{1 0 ^{n}}{9 ^{n}}

x \frac{d y}{d x} - y = x^{2} e^{x}

\frac{\partial}{\partial x} (ln (3 x + 5 y))

x \to π lim ((2 x - 2 π) cot (x))

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