integral of (32^x+43^x)/(2^x)

解答

\int \frac{3 \cdot 2 ^{x} + 4 \cdot 3 ^{x}}{2 ^{x}} d x

3 x + \frac{3 ^{x} \cdot 2 ^{2 - x}}{ln ( \frac{3}{2} )} + C

求解步骤

\int \frac{3 \cdot 2 ^{x} + 4 \cdot 3 ^{x}}{2 ^{x}} d x

\frac{3 \cdot 2 ^{x} + 4 \cdot 3 ^{x}}{2 ^{x}} = (\frac{3 \cdot 2 ^{x}}{2 ^{x}} + \frac{4 \cdot 3 ^{x}}{2 ^{x}})

= \int \frac{3 \cdot 2 ^{x}}{2 ^{x}} + \frac{4 \cdot 3 ^{x}}{2 ^{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 \frac{3 \cdot 2 ^{x}}{2 ^{x}} d x + \int \frac{4 \cdot 3 ^{x}}{2 ^{x}} d x

\int \frac{3 \cdot 2 ^{x}}{2 ^{x}} d x = 3 x

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

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

解答补常数

= 3 x + \frac{3 ^{x} \cdot 2 ^{2 - x}}{ln ( \frac{3}{2} )} + C

\int 15 ln (x^{2} - x + 12) d x

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