integral from 0 to 1 of x^2sin(x^3)

解答

\int_{0}^{1} x^{2} sin (x^{3}) d x

\frac{1}{3} (- cos (1) + 1)

十进制

0.15323 \dots

求解步骤

\int_{0}^{1} x^{2} sin (x^{3}) d x

使用换元积分法

= \int_{0}^{1} \frac{sin ( u )}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int_{0}^{1} sin (u) d u

使用积分常见定则:

\int sin (u) d u = - cos (u)

= \frac{1}{3} [- cos (u)]_{0}^{1}

带入上下限计算后得:

- cos (1) + 1

= \frac{1}{3} (- cos (1) + 1)

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