integral of (e^{2x}-1)/(e^{2x)+3}

解答

\int \frac{e ^{2 x} - 1}{e ^{2 x} + 3} d x

- \frac{1}{3} x + \frac{2}{3} ln e^{2 x} + 3 + C

求解步骤

\int \frac{e ^{2 x} - 1}{e ^{2 x} + 3} d x

\frac{e ^{2 x} - 1}{e ^{2 x} + 3} = - \frac{4}{e ^{2 x} + 3} + 1

= \int - \frac{4}{e ^{2 x} + 3} + 1 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{4}{e ^{2 x} + 3} d x + \int 1 d x

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

\int 1 d x = x

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

化简

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

= - \frac{1}{3} x + \frac{2}{3} ln e^{2 x} + 3

解答补常数

= - \frac{1}{3} x + \frac{2}{3} ln e^{2 x} + 3 + C

x \to 3 lim (\frac{x + 1}{x ^{2} - 9})

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\frac{d}{d x} (- \frac{1}{64} sin (\frac{x}{8}))

\frac{d}{d t} (a cos (wt) + b sin (wt))

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