integral of (2x-3)/(11-4x^2)

解答

\int \frac{2 x - 3}{11 - 4 x ^{2}} d x

- \frac{1}{4} ln 11 - 4 x^{2} - \frac{3}{4 11} (ln \frac{2}{11} x + 1 - ln \frac{2}{11} x - 1) + C

求解步骤

\int \frac{2 x - 3}{11 - 4 x ^{2}} d x

乘开

\frac{2 x - 3}{11 - 4 x ^{2}} : \frac{2 x}{11 - 4 x ^{2}} - \frac{3}{11 - 4 x ^{2}}

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{2 x}{11 - 4 x ^{2}} d x - \int \frac{3}{11 - 4 x ^{2}} d x

\int \frac{2 x}{11 - 4 x ^{2}} d x = - \frac{1}{4} ln 11 - 4 x^{2}

\int \frac{3}{11 - 4 x ^{2}} d x = \frac{3}{4 11} (ln \frac{2}{11} x + 1 - ln \frac{2}{11} x - 1)

= - \frac{1}{4} ln 11 - 4 x^{2} - \frac{3}{4 11} (ln \frac{2}{11} x + 1 - ln \frac{2}{11} x - 1)

解答补常数

= - \frac{1}{4} ln 11 - 4 x^{2} - \frac{3}{4 11} (ln \frac{2}{11} x + 1 - ln \frac{2}{11} x - 1) + C

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