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

解答

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

\frac{9}{10} ln ∣ x - 3 ∣ + \frac{3}{20} ln x^{2} + 1 + \frac{9}{10} arctan (x) + \frac{1}{5} (- 3 ln x^{2} + 1 + 2 arctan (x)) + C

求解步骤

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

乘开

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

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

\int \frac{4 x}{( x - 3 ) ( x ^{2} + 1 )} d x = 4 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x)))

\int \frac{3}{( x - 3 ) ( x ^{2} + 1 )} d x = 3 (\frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)))

= 4 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x))) - 3 (\frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)))

化简

4 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{3}{2} ln x^{2} + 1 + arctan (x))) - 3 (\frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x))) : \frac{9}{10} ln ∣ x - 3 ∣ + \frac{3}{20} ln x^{2} + 1 + \frac{9}{10} arctan (x) + \frac{1}{5} (- 3 ln x^{2} + 1 + 2 arctan (x))

= \frac{9}{10} ln ∣ x - 3 ∣ + \frac{3}{20} ln x^{2} + 1 + \frac{9}{10} arctan (x) + \frac{1}{5} (- 3 ln x^{2} + 1 + 2 arctan (x))

解答补常数

= \frac{9}{10} ln ∣ x - 3 ∣ + \frac{3}{20} ln x^{2} + 1 + \frac{9}{10} arctan (x) + \frac{1}{5} (- 3 ln x^{2} + 1 + 2 arctan (x)) + C

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