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受欢迎的微积分 >

积分 1/((x-2)^3(x+4)^4)

解答

积分

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

解答

\frac{5}{23328} ln ∣ x - 2 ∣ + \frac{1}{1944 ( x - 2 )} - \frac{1}{2592 ( x - 2 ) ^{2}} - \frac{5}{23328} ln ∣ x + 4 ∣ + \frac{1}{1296 ( x + 4 )} + \frac{1}{864 ( x + 4 ) ^{2}} + \frac{1}{648 ( x + 4 ) ^{3}} + C

求解步骤

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

将

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

用部份分式展开:

\frac{5}{23328 ( x - 2 )} - \frac{1}{1944 ( x - 2 ) ^{2}} + \frac{1}{1296 ( x - 2 ) ^{3}} - \frac{5}{23328 ( x + 4 )} - \frac{1}{1296 ( x + 4 ) ^{2}} - \frac{1}{432 ( x + 4 ) ^{3}} - \frac{1}{216 ( x + 4 ) ^{4}}

= \int \frac{5}{23328 ( x - 2 )} - \frac{1}{1944 ( x - 2 ) ^{2}} + \frac{1}{1296 ( x - 2 ) ^{3}} - \frac{5}{23328 ( x + 4 )} - \frac{1}{1296 ( x + 4 ) ^{2}} - \frac{1}{432 ( x + 4 ) ^{3}} - \frac{1}{216 ( x + 4 ) ^{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{5}{23328 ( x - 2 )} d x - \int \frac{1}{1944 ( x - 2 ) ^{2}} d x + \int \frac{1}{1296 ( x - 2 ) ^{3}} d x - \int \frac{5}{23328 ( x + 4 )} d x - \int \frac{1}{1296 ( x + 4 ) ^{2}} d x - \int \frac{1}{432 ( x + 4 ) ^{3}} d x - \int \frac{1}{216 ( x + 4 ) ^{4}} d x

\int \frac{5}{23328 ( x - 2 )} d x = \frac{5}{23328} ln ∣ x - 2 ∣

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

\int \frac{1}{1296 ( x - 2 ) ^{3}} d x = - \frac{1}{2592 ( x - 2 ) ^{2}}

\int \frac{5}{23328 ( x + 4 )} d x = \frac{5}{23328} ln ∣ x + 4 ∣

\int \frac{1}{1296 ( x + 4 ) ^{2}} d x = - \frac{1}{1296 ( x + 4 )}

\int \frac{1}{432 ( x + 4 ) ^{3}} d x = - \frac{1}{864 ( x + 4 ) ^{2}}

\int \frac{1}{216 ( x + 4 ) ^{4}} d x = - \frac{1}{648 ( x + 4 ) ^{3}}

= \frac{5}{23328} ln ∣ x - 2 ∣ - (- \frac{1}{1944 ( x - 2 )}) - \frac{1}{2592 ( x - 2 ) ^{2}} - \frac{5}{23328} ln ∣ x + 4 ∣ - (- \frac{1}{1296 ( x + 4 )}) - (- \frac{1}{864 ( x + 4 ) ^{2}}) - (- \frac{1}{648 ( x + 4 ) ^{3}})

化简

= \frac{5}{23328} ln ∣ x - 2 ∣ + \frac{1}{1944 ( x - 2 )} - \frac{1}{2592 ( x - 2 ) ^{2}} - \frac{5}{23328} ln ∣ x + 4 ∣ + \frac{1}{1296 ( x + 4 )} + \frac{1}{864 ( x + 4 ) ^{2}} + \frac{1}{648 ( x + 4 ) ^{3}}

解答补常数

= \frac{5}{23328} ln ∣ x - 2 ∣ + \frac{1}{1944 ( x - 2 )} - \frac{1}{2592 ( x - 2 ) ^{2}} - \frac{5}{23328} ln ∣ x + 4 ∣ + \frac{1}{1296 ( x + 4 )} + \frac{1}{864 ( x + 4 ) ^{2}} + \frac{1}{648 ( x + 4 ) ^{3}} + C

作图

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