integral of (x^2+1)/((x-10)(x-9)^2)

解答

\int \frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}} d x

101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9} + C

求解步骤

\int \frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}} d x

将

\frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}}

用部份分式展开:

\frac{101}{x - 10} - \frac{100}{x - 9} - \frac{82}{( x - 9 ) ^{2}}

= \int \frac{101}{x - 10} - \frac{100}{x - 9} - \frac{82}{( x - 9 ) ^{2}} 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{101}{x - 10} d x - \int \frac{100}{x - 9} d x - \int \frac{82}{( x - 9 ) ^{2}} d x

\int \frac{101}{x - 10} d x = 101 ln ∣ x - 10 ∣

\int \frac{100}{x - 9} d x = 100 ln ∣ x - 9 ∣

\int \frac{82}{( x - 9 ) ^{2}} d x = - \frac{82}{x - 9}

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ - (- \frac{82}{x - 9})

化简

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9}

解答补常数

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9} + C

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