integral of (3^{5x}-1)/(3^{5x)}

解答

\int \frac{3 ^{5 x} - 1}{3 ^{5 x}} d x

x + \frac{1}{5 ln ( 3 ) \cdot 24 3 ^{x}} + C

求解步骤

\int \frac{3 ^{5 x} - 1}{3 ^{5 x}} d x

化简

\frac{3 ^{5 x} - 1}{3 ^{5 x}} : \frac{24 3 ^{x} - 1}{24 3 ^{x}}

= \int \frac{24 3 ^{x} - 1}{24 3 ^{x}} d x

乘开

\frac{24 3 ^{x} - 1}{24 3 ^{x}} : 1 - \frac{1}{24 3 ^{x}}

= \int 1 - \frac{1}{24 3 ^{x}} 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 1 d x - \int \frac{1}{24 3 ^{x}} d x

\int 1 d x = x

\int \frac{1}{24 3 ^{x}} d x = - \frac{1}{5 ln ( 3 ) \cdot 24 3 ^{x}}

= x - (- \frac{1}{5 ln ( 3 ) \cdot 24 3 ^{x}})

化简

= x + \frac{1}{5 ln ( 3 ) \cdot 24 3 ^{x}}

解答补常数

= x + \frac{1}{5 ln ( 3 ) \cdot 24 3 ^{x}} + C

d er i v a t i v e 3 (5 e^{5 x} cos (6 x) - 6 e^{5 x} sin (6 x))

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