integral of (2x+7)/((2x^2+14x)^4)

解答

\int \frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}} d x

- \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}} + C

求解步骤

\int \frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}} d x

将

\frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}}

用部份分式展开:

\frac{1}{134456 x ^{2}} - \frac{1}{19208 x ^{3}} + \frac{1}{5488 x ^{4}} - \frac{1}{134456 ( x + 7 ) ^{2}} - \frac{1}{19208 ( x + 7 ) ^{3}} - \frac{1}{5488 ( x + 7 ) ^{4}}

= \int \frac{1}{134456 x ^{2}} - \frac{1}{19208 x ^{3}} + \frac{1}{5488 x ^{4}} - \frac{1}{134456 ( x + 7 ) ^{2}} - \frac{1}{19208 ( x + 7 ) ^{3}} - \frac{1}{5488 ( 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{1}{134456 x ^{2}} d x - \int \frac{1}{19208 x ^{3}} d x + \int \frac{1}{5488 x ^{4}} d x - \int \frac{1}{134456 ( x + 7 ) ^{2}} d x - \int \frac{1}{19208 ( x + 7 ) ^{3}} d x - \int \frac{1}{5488 ( x + 7 ) ^{4}} d x

\int \frac{1}{134456 x ^{2}} d x = - \frac{1}{134456 x}

\int \frac{1}{19208 x ^{3}} d x = - \frac{1}{38416 x ^{2}}

\int \frac{1}{5488 x ^{4}} d x = - \frac{1}{16464 x ^{3}}

\int \frac{1}{134456 ( x + 7 ) ^{2}} d x = - \frac{1}{134456 ( x + 7 )}

\int \frac{1}{19208 ( x + 7 ) ^{3}} d x = - \frac{1}{38416 ( x + 7 ) ^{2}}

\int \frac{1}{5488 ( x + 7 ) ^{4}} d x = - \frac{1}{16464 ( x + 7 ) ^{3}}

= - \frac{1}{134456 x} - (- \frac{1}{38416 x ^{2}}) - \frac{1}{16464 x ^{3}} - (- \frac{1}{134456 ( x + 7 )}) - (- \frac{1}{38416 ( x + 7 ) ^{2}}) - (- \frac{1}{16464 ( x + 7 ) ^{3}})

化简

= - \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}}

解答补常数

= - \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}} + C

\int sin (x) (1 + cos (x))^{4} d x

\frac{\partial}{\partial y} (x^{2} + y^{2} - z^{2})

l a pl a ce t r an s f or m e^{- t} (\frac{e ^{t} + e ^{- t}}{2})

t an g e n t f (x) = \frac{25 x ^{3}}{3} + 10 x^{2} + 4 x - 7

n \to \infty lim (4 n)