integral of (x^3)/(sqrt(1-x^8))

解答

\int \frac{x ^{3}}{1 - x ^{8}} d x

\frac{1}{4} arcsin (x^{4}) + C

求解步骤

\int \frac{x ^{3}}{1 - x ^{8}} d x

使用换元积分法

= \int \frac{1}{4 1 - u ^{2}} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{1}{1 - u ^{2}} d u

使用积分常见定则:

\int \frac{1}{1 - u ^{2}} d u = arcsin (u)

= \frac{1}{4} arcsin (u)

u = x^{4}

代回

= \frac{1}{4} arcsin (x^{4})

解答补常数

= \frac{1}{4} arcsin (x^{4}) + C

d er i v a t i v e f (x) = 3 - 4 x

d er i v a t i v e y = 4 cos (x)

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

\int_{1}^{10} x ln (x) d x

x \to \infty lim (3 x - 9 x^{2} + x)