integral of (x-4)/(x^2+2x-15)

解答

\int \frac{x - 4}{x ^{2} + 2 x - 15} d x

\frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) + \frac{1}{2} ln \frac{1}{4} (x + 1) + 1 - \frac{1}{2} ln \frac{1}{4} (x + 1) - 1 + C

求解步骤

\int \frac{x - 4}{x ^{2} + 2 x - 15} d x

乘开

\frac{x - 4}{x ^{2} + 2 x - 15} : \frac{x}{x ^{2} + 2 x - 15} - \frac{4}{x ^{2} + 2 x - 15}

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{x}{x ^{2} + 2 x - 15} d x - \int \frac{4}{x ^{2} + 2 x - 15} d x

\int \frac{x}{x ^{2} + 2 x - 15} d x = \frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1)

\int \frac{4}{x ^{2} + 2 x - 15} d x = - \frac{1}{2} ln \frac{x + 1}{4} + 1 + \frac{1}{2} ln \frac{x + 1}{4} - 1

= \frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) - (- \frac{1}{2} ln \frac{x + 1}{4} + 1 + \frac{1}{2} ln \frac{x + 1}{4} - 1)

化简

\frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) - (- \frac{1}{2} ln \frac{x + 1}{4} + 1 + \frac{1}{2} ln \frac{x + 1}{4} - 1) : \frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) + \frac{1}{2} ln \frac{1}{4} (x + 1) + 1 - \frac{1}{2} ln \frac{1}{4} (x + 1) - 1

= \frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) + \frac{1}{2} ln \frac{1}{4} (x + 1) + 1 - \frac{1}{2} ln \frac{1}{4} (x + 1) - 1

解答补常数

= \frac{1}{2} ln x^{2} + 2 x - 15 + \frac{1}{8} (ln \frac{1}{4} (x + 1) + 1 - ln \frac{1}{4} (x + 1) - 1) + \frac{1}{2} ln \frac{1}{4} (x + 1) + 1 - \frac{1}{2} ln \frac{1}{4} (x + 1) - 1 + C

\int_{0}^{1} \frac{x}{1 + x ^{4}} d x

a re a y = 3 x, [3, 7]

\int 34 arctan (x) d x

x \to 2 lim ((x + 2) \cdot (2 x - 1))

t a y l or ln (x + 1), 1