integral of 5^{2-3x}

解答

\int 5^{2 - 3 x} d x

- \frac{5 ^{2 - 3 x}}{3 ln ( 5 )} + C

求解步骤

\int 5^{2 - 3 x} d x

使用换元积分法

= \int - \frac{1}{3} \cdot 5^{u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{3} \cdot \int 5^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

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

u = 2 - 3 x

代回

= - \frac{1}{3} \cdot \frac{5 ^{2 - 3 x}}{ln ( 5 )}

化简

- \frac{1}{3} \cdot \frac{5 ^{2 - 3 x}}{ln ( 5 )} : - \frac{5 ^{2 - 3 x}}{3 ln ( 5 )}

= - \frac{5 ^{2 - 3 x}}{3 ln ( 5 )}

解答补常数

= - \frac{5 ^{2 - 3 x}}{3 ln ( 5 )} + C

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