integral of (x^e+e^{2x}+x^pi+pi^x+1)

解答

\int (x^{e} + e^{2 x} + x^{π} + π^{x} + 1) d x

\frac{x ^{e + 1}}{e + 1} + \frac{1}{2} e^{2 x} + \frac{x ^{π + 1}}{π + 1} + \frac{π ^{x}}{ln ( π )} + x + C

求解步骤

\int x^{e} + e^{2 x} + x^{π} + π^{x} + 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 x^{e} d x + \int e^{2 x} d x + \int x^{π} d x + \int π^{x} d x + \int 1 d x

\int x^{e} d x = \frac{x ^{e + 1}}{e + 1}

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

\int x^{π} d x = \frac{x ^{π + 1}}{π + 1}

\int π^{x} d x = \frac{π ^{x}}{ln ( π )}

\int 1 d x = x

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

解答补常数

= \frac{x ^{e + 1}}{e + 1} + \frac{1}{2} e^{2 x} + \frac{x ^{π + 1}}{π + 1} + \frac{π ^{x}}{ln ( π )} + x + C

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