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受欢迎的微积分 >

integral from 0 to 20 of 2^{t/(10)}

解答

\int_{0}^{20} 2^{\frac{t}{10}} d t

解答

\frac{30}{ln ( 2 )}

+1

十进制

43.28085 \dots

求解步骤

\int_{0}^{20} 2^{\frac{t}{10}} d t

使用换元积分法

= \int_{0}^{2} 5 \cdot 2^{u + 1} d u

提出常数:

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

= 5 \cdot \int_{0}^{2} 2^{u + 1} d u

2^{u + 1} = 2^{1} \cdot 2^{u}

= 5 \cdot \int_{0}^{2} 2^{1} \cdot 2^{u} d u

提出常数:

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

= 5 \cdot 2^{1} \cdot \int_{0}^{2} 2^{u} d u

化简

= 5 \cdot 2 \cdot \int_{0}^{2} 2^{u} d u

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

= 5 \cdot 2 [\frac{2 ^{u}}{ln ( 2 )}]_{0}^{2}

化简

= 10 [\frac{2 ^{u}}{ln ( 2 )}]_{0}^{2}

带入上下限计算后得:

\frac{3}{ln ( 2 )}

= 10 \cdot \frac{3}{ln ( 2 )}

化简

= \frac{30}{ln ( 2 )}

作图

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