integral of 1/((4-x^2)^3)

解答

\int \frac{1}{( 4 - x ^{2} ) ^{3}} d x

\frac{3}{512} ln ∣ x + 2 ∣ - \frac{3}{256 ( x + 2 )} - \frac{1}{128 ( x + 2 ) ^{2}} - \frac{3}{512} ln ∣ x - 2 ∣ - \frac{3}{256 ( x - 2 )} + \frac{1}{128 ( x - 2 ) ^{2}} + C

求解步骤

\int \frac{1}{( 4 - x ^{2} ) ^{3}} d x

将

\frac{1}{( 4 - x ^{2} ) ^{3}}

用部份分式展开:

\frac{3}{512 ( x + 2 )} + \frac{3}{256 ( x + 2 ) ^{2}} + \frac{1}{64 ( x + 2 ) ^{3}} - \frac{3}{512 ( x - 2 )} + \frac{3}{256 ( x - 2 ) ^{2}} - \frac{1}{64 ( x - 2 ) ^{3}}

= \int \frac{3}{512 ( x + 2 )} + \frac{3}{256 ( x + 2 ) ^{2}} + \frac{1}{64 ( x + 2 ) ^{3}} - \frac{3}{512 ( x - 2 )} + \frac{3}{256 ( x - 2 ) ^{2}} - \frac{1}{64 ( x - 2 ) ^{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{3}{512 ( x + 2 )} d x + \int \frac{3}{256 ( x + 2 ) ^{2}} d x + \int \frac{1}{64 ( x + 2 ) ^{3}} d x - \int \frac{3}{512 ( x - 2 )} d x + \int \frac{3}{256 ( x - 2 ) ^{2}} d x - \int \frac{1}{64 ( x - 2 ) ^{3}} d x

\int \frac{3}{512 ( x + 2 )} d x = \frac{3}{512} ln ∣ x + 2 ∣

\int \frac{3}{256 ( x + 2 ) ^{2}} d x = - \frac{3}{256 ( x + 2 )}

\int \frac{1}{64 ( x + 2 ) ^{3}} d x = - \frac{1}{128 ( x + 2 ) ^{2}}

\int \frac{3}{512 ( x - 2 )} d x = \frac{3}{512} ln ∣ x - 2 ∣

\int \frac{3}{256 ( x - 2 ) ^{2}} d x = - \frac{3}{256 ( x - 2 )}

\int \frac{1}{64 ( x - 2 ) ^{3}} d x = - \frac{1}{128 ( x - 2 ) ^{2}}

= \frac{3}{512} ln ∣ x + 2 ∣ - \frac{3}{256 ( x + 2 )} - \frac{1}{128 ( x + 2 ) ^{2}} - \frac{3}{512} ln ∣ x - 2 ∣ - \frac{3}{256 ( x - 2 )} - (- \frac{1}{128 ( x - 2 ) ^{2}})

化简

= \frac{3}{512} ln ∣ x + 2 ∣ - \frac{3}{256 ( x + 2 )} - \frac{1}{128 ( x + 2 ) ^{2}} - \frac{3}{512} ln ∣ x - 2 ∣ - \frac{3}{256 ( x - 2 )} + \frac{1}{128 ( x - 2 ) ^{2}}

解答补常数

= \frac{3}{512} ln ∣ x + 2 ∣ - \frac{3}{256 ( x + 2 )} - \frac{1}{128 ( x + 2 ) ^{2}} - \frac{3}{512} ln ∣ x - 2 ∣ - \frac{3}{256 ( x - 2 )} + \frac{1}{128 ( x - 2 ) ^{2}} + C

\int ln (x + 1) x d x

\frac{\partial}{\partial v} (\frac{u}{u + v})

y^{^{'}} = e^{6 x} - 4 x

t an g e n t f (x) = x^{2} - 3 x + 1, at x = 1

\frac{\partial}{\partial x} (x^{2} - 4 x y + y^{3} + 4 y)