integral from 0 to 4 of pi(16x^2-x^4)

解答

\int_{0}^{4} π (16 x^{2} - x^{4}) d x

π \frac{2048}{15}

十进制

428.93211 \dots

求解步骤

\int_{0}^{4} π (16 x^{2} - x^{4}) d x

提出常数:

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

= π \cdot \int_{0}^{4} 16 x^{2} - x^{4} 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

= π (\int_{0}^{4} 16 x^{2} d x - \int_{0}^{4} x^{4} d x)

\int_{0}^{4} 16 x^{2} d x = \frac{1024}{3}

\int_{0}^{4} x^{4} d x = \frac{1024}{5}

= π (\frac{1024}{3} - \frac{1024}{5})

化简

\frac{1024}{3} - \frac{1024}{5} : \frac{2048}{15}

= π \frac{2048}{15}

\int \frac{2 x}{1 - 8 x ^{2}} d x

\int \frac{e ^{7 t}}{t} d t

x \to 5 lim (\frac{3 x ^{2} - 13 x - 10}{2 x ^{2} - 7 x - 15})

\frac{d}{d x} (\frac{x ^{2} - 3 x}{x + 1})

\int 9 x e^{17 x} d x