面积 (4/x),(16x),(1/4 x)

解答

面积

(\frac{4}{x}), (16 x), (\frac{1}{4} x)

24 ln (2)

十进制

16.63553 \dots

求解步骤

要找到相交点，解

f_{i} (x) = f_{j} (x)

(\frac{4}{x}) = (16 x) : x = \frac{1}{2}, x = - \frac{1}{2}

(\frac{4}{x}) = (\frac{1}{4} x) : x = 4, x = - 4

(16 x) = (\frac{1}{4} x) : x = 0

= \int_{- 4}^{- \frac{1}{2}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{1}{2}}^{0} ∣ f_{3} (x) - f_{2} (x) ∣ d x + \int_{0}^{\frac{1}{2}} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{\frac{1}{2}}^{4} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- 4}^{- \frac{1}{2}} (\frac{4}{x}) - (\frac{1}{4} x) d x + \int_{- \frac{1}{2}}^{0} (\frac{1}{4} x) - (16 x) d x + \int_{0}^{\frac{1}{2}} (16 x) - (\frac{1}{4} x) d x + \int_{\frac{1}{2}}^{4} (\frac{4}{x}) - (\frac{1}{4} x) d x

\int_{- 4}^{- \frac{1}{2}} (\frac{4}{x}) - (\frac{1}{4} x) d x = 12 ln (2) - \frac{63}{32}

\int_{- \frac{1}{2}}^{0} (\frac{1}{4} x) - (16 x) d x = \frac{63}{32}

\int_{0}^{\frac{1}{2}} (16 x) - (\frac{1}{4} x) d x = \frac{63}{32}

\int_{\frac{1}{2}}^{4} (\frac{4}{x}) - (\frac{1}{4} x) d x = 12 ln (2) - \frac{63}{32}

= 12 ln (2) - \frac{63}{32} + \frac{63}{32} + \frac{63}{32} + 12 ln (2) - \frac{63}{32}

化简

= 24 ln (2)

x \to 2 + lim (\frac{2}{x ^{2} - 4})

\int \frac{3 ^{5 x} - 1}{3 ^{5 x}} d x

d er i v a t i v e 3 (5 e^{5 x} cos (6 x) - 6 e^{5 x} sin (6 x))

n = 0 \sum \infty (- \frac{1}{6})^{n}

\int \frac{1}{x ^{5} 4 x ^{2} - 1} d x