integral from 0 to 2 of 5^{2x}

解答

\int_{0}^{2} 5^{2 x} d x

\frac{312}{ln ( 5 )}

十进制

193.85649 \dots

求解步骤

\int_{0}^{2} 5^{2 x} d x

化简

5^{2 x} : 2 5^{x}

= \int_{0}^{2} 2 5^{x} d x

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

= [\frac{2 5 ^{x}}{ln ( 25 )}]_{0}^{2}

化简

ln (25) : 2 ln (5)

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

带入上下限计算后得:

\frac{312}{ln ( 5 )}

= \frac{312}{ln ( 5 )}

\int_{1}^{2} cos (2 θ) cos (3 θ) d θ

\int_{1}^{4} x^{2} - 2 x - 3 d x

\int_{0}^{a} 1 - \frac{x ^{2}}{a ^{2}} d x

\int_{0}^{2} π (4 y^{2} - y^{4}) d y

\int_{- 1}^{1} \frac{1}{1 - x} d x